# Integral transforms related to Nevanlinna-Pick functions from an analytic,
probabilistic and free-probability point of view

Wissem Jedidi Department of Statistics & OR, King Saud University, P.O. Box
2455, Riyadh 11451, Saudi Arabia, , Université de Tunis El Manar, Faculté des
Sciences de Tunis, Département de Mathématiques, Laboratoire d’Analyse
Mathématiques et Applications LR11ES11. 2092 - El Manar I, Tunis, Tunisia,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Zbigniew
J. Jurek and Nuha Taymani ∗ University of Wrocław, Poland,
<EMAIL_ADDRESS>
###### Abstract

We establish a new connection between the class of Nevanlinna-Pick functions
and the one of the exponents associated to spectrally negative Lévy processes.
As a consequence, we compute the characteristics related to some hyperbolic
functions and we show a property of temporal complete monotonicity, similar to
the one obtained via the Lamperti transformation by Bertoin & Yor (On
subordinators, self-similar Markov processes and some factorizations of the
exponential variable, Elect. Comm. in Probab., vol. 6, pp. 95–106, 2001) for
self-similar Markov processes. More precisely, we show the remarkable fact
that for a subordinator $\xi$, the function $t\mapsto
t^{n}\,\mathbb{E}[\xi_{t}^{-p}]$ is , depending on the values of the exponents
$n=0,1,2,\;p>-1$, or a Bernstein function or a completely monotone function.
In particular, $\xi$ is the inverse time subordinator of a spectrally negative
Lévy process, if, and only if, for some $\,p\geq 1$, the function $t\mapsto
t\,\mathbb{E}[\xi_{t}^{-p}]$ is a Stieltjes transform. Finally, we clarify to
which extent Nevanlinna-Pick functions are related to free-probability and to
Voiculescu transforms, and we provide an inversion procedure.

###### Keywords:

Characteristic functions; Laplace Transform; Cauchy transform; Voiculescu
transform; Convolution; Infinite divisibility; Free-infinite divisibility;
Lévy processes; Subordinators; Analytic functions; Nevanlinna-Pick functions;
Bernstein functions; Complete Bernstein functions; Hyperbolic characteristic
functions.

August 27, 2024

##  1 Introduction

In complex analysis, analytic functions that preserve the upper complex half-
plane play an important role. According to Nevanlinna-Pick theorem , these are
precisely functions $F$ that admit the (unique) representation:

$F(z)=\mathrm{a}+\mathrm{b}\;z+\int_{\mathds{R}}\frac{zx-1}{z+x}\rho(dx),\quad\mbox{for}\;z\in\mathds{C}\setminus\mathds{R},\;\mbox{where}\;\mathrm{a}\in\mathds{R},\,\mathrm{b}\geq
0,\;\mbox{and $\rho$ is a finite Borel measure.}$ (1)

We will denote by $\mathcal{N}$ the set of all Nevanlinna-Pick functions while
$\mathcal{N}(A)$, for a Borel set $A$, stands for

$\mathcal{N}(A):=\\{\,F\in\mathcal{N}\;\mbox{ such that in
\eqref{n1}},\;support(\rho)\subset A\\}.$

On the other hand, there are complete Bernstein functions, cf. Appendix, i.e.
those functions $f:(0,\infty)\to\mathds{R}$ that analytically extend to
$z\in\mathds{C}\setminus(-\infty,0]$ and have a form as Nevanlinna-Pick
functions as in (1). Furthermore, in the free-probability theory (cf.
Appendix) an infinitely divisible probability measure $\nu$ has its Voiculescu
transform $V_{\nu}(z)$ as follows

$V_{\nu}(z)=\mathrm{a}+\int_{\mathds{R}}\frac{zx-1}{z+x}m(dx),\quad\mbox{for}\;z\in\mathds{C}\setminus\mathds{R},$
(2)

for a uniquely determined finite measure $m$. In this paper, we will not use
the analyticity property of Nevanlinna-Pick functions or Voiculesu transforms.
In fact, from the values of $F(it)$ or $V_{\nu}(it),\;t\in\mathds{R}$, we can
retrieve the representing parameters, in particular, the $\rho$-measures, cf.
(7) below.

Recall that in the (classical) probability we have that a measure $\mu$ is
infinitely divisible if and only if its Fourier transform (or characteristic
function) $\mathfrak{F}_{\mu}(t)$ is of a form

$\mathfrak{F}_{\mu}(s):=\int_{\mathds{R}}e^{i\,s\,x}\mu(dx)=\exp\Big{[}i\,u\,\mathrm{a}-\mathrm{b}^{2}u^{2}+\int_{\mathds{R}}\left(e^{i\,s\,x}-1-\frac{itx}{1+x^{2}}\right)\frac{1+x^{2}}{x^{2}}m(dx)\Big{]},$
(3)

where $\mathrm{a}\in\mathds{R},$ $\mathrm{b}\geq 0,$ and $m$ is a finite Borel
measure. The above is the famous Lévy-Khintchine formula for infinitely
divisible characteristic functions and those are closely related to Lévy
stochastic process. Note that (2) and (3) give a bijection between the
classical infinite divisibility and the free-probability infinite
divisibility. Namely, $\nu$ in (2) is a free-probability analogue of $\mu$ in
the classical probability (when $\mathrm{b}=0$).

In this paper, Theorem 2.1, gives some equivalent characterizations of
$F\in\mathcal{N}(A)$, among them, there is a way to associate $F$ with a
Cauchy transform. In Proposition and Theorem 3.1, we establish a temporal
monotonicity property for subordinators. Finally, in Propositions 3 and 5, are
found Voiculescu transforms of free-probability analogues of the classical
infinitely divisible hyperbolic functions. In particular, the $\rho$-measures
in (2) are given as a composition of Laplace and Fourier transforms.

In order to precise our main results, we need to formalize our objects. The
right, left, upper and lower complex half-plane are denoted by

$\mathds{C}^{+}:=\\{z\in\mathds{C}:\Re(z)>0\\},\quad\mathds{C}^{-}:=-\mathds{C}^{+},\quad\mathds{H}^{+}:=\\{z\in\mathds{C}:\Im(z)>0\\},\quad\mathds{H}^{-}:=-\mathds{H}^{+}.$

The Alexandrov compactification of $\mathds{C}\cup\\{\infty\\}$ is denoted by
$\overline{\mathds{C}}$. For given $\alpha,N>0$ and $M\in\mathds{R}$, we
introduce the angular sectors and cones in $\mathds{H}^{+}$:

$\Gamma_{\alpha}(M):=\\{z=a+ib\in\mathds{H}^{+}\;:\;\alpha
b>|a-M|\\},\qquad\Gamma_{\alpha}:=\Gamma_{\alpha}(0)=\Gamma_{\alpha}(M)-M,\qquad\Gamma_{\alpha,N}:=\\{z=a+ib\in\Gamma_{\alpha}\;:\;b>N\\}.$
(4)

In classical complex analysis, one of the fundamental results is the integral
representation of _analytic functions_ defined on $\mathds{H}^{+}$, or
$\mathds{C}{\footnotesize\setminus}\mathds{R}=\mathds{H}^{+}\cup\mathds{H}^{-}$,
that preserve $\mathds{H}^{+}$. If such an analytic function, say $F$, is only
defined on $\mathds{H}^{+}$, then we can extend it onto
$\mathds{C}{\footnotesize\setminus}\mathds{R}$ by setting
$F(z)=\overline{F(\overline{z})}$, if $z\in\mathds{H}^{-}$. Then, $F$ admits
the (unique) canonical form: for
$z\in\mathds{C}{\footnotesize\setminus}\mathds{R}$,

$\displaystyle F(z)$ $\displaystyle=$
$\displaystyle\mathrm{a}+\mathrm{b}z+\int_{\mathds{R}}\frac{zx-1}{z+x}\rho(dx)$
(5) $\displaystyle=$
$\displaystyle\mathrm{a}+\mathrm{b}z+\int_{\mathds{R}}\left(\frac{x}{1+x^{2}}-\frac{1}{x+z}\right)(1+x^{2})\rho(dx).$
(6)

where $\mathrm{a}\in\mathds{R}$ is a real number, $\mathrm{b}\geq 0$ and
$\rho$ is a finite (Borel) measure on $\mathds{R}$. Note that

$\mathrm{a}=\Re\left(F(i)\right),\quad\mathrm{b}=\lim_{u\to\infty}F(iu)/(iu)\quad\mbox{and}\quad\rho(\mathds{R})=\Im\left(F(i)\right)-\mathrm{b},$
(7)

that for every measure $\rho$ we have the inversion formula, cf. (Akhi, ,
p.126) or Lang (Lang, , p.380):

$\int_{(u,v]}(1+x^{2})\rho(dx)=\lim_{\epsilon\to
0^{+}}\frac{1}{\pi}\int_{(u,v]}\Im\left(F(-x+i\epsilon)\right)\,dx,\quad\mbox{whenever
$\rho(\\{u,v\\})=0$},$ (8)

and that the half-plane preservation property could be seen from property (P1)
in Section 2 , of the function $f$ in (17) below. Those function $F$ are
coined in the literature as Pick or Nevanlinna-Pick functions. cf. Akhiezer
Akhi , Bondesson (Bon, , p.21), or Schilling, Song and Vondraček (SSV, ,
Chapter 6). The (unique) representing measure $\rho$ appearing in (5) and (6)
is called the Nevanlinna measure.

In a couple of last decades, representations of the form (5) or (6), with
$\mathrm{b}=0$, appeared in the _free-probability_ as the free-probability
analog of the classical Lévy-Khintchine formula for infinitely divisible
characteristic functions (Fourier transforms). More precisely, they are
defined as

$V(z):=\mathrm{a}+\int_{\mathds{R}}\frac{1-zx}{z+x}\rho(dx),\quad
z\in\mathds{C}\setminus\mathds{R}$ (9)

and are called Voiculescu transforms, cf. Bercovici and Voiculescu (bv, ,
Section 5, in particular Theorem 5.10). Observe that if
$support(\rho)=A\subset\mathds{R}$, then

  * •

$-V\in\mathcal{N}(A)$;

  * •

if $\widehat{\rho}$ is the image of $\rho$ by the mapping $x\mapsto-x$, then

$z\mapsto
V(-z):=\mathrm{a}+\int_{-A}\frac{zx-1}{z+x}\widehat{\rho}(dx)\in\mathcal{N}(-A);$

  * •

if $\check{A}$ and $\check{\rho}$ are the images of $A\setminus\\{0\\}$ and
$\rho_{A\setminus\\{0\\}}$ by the mapping $x\mapsto 1/x$, then

$z\mapsto
V\left(\frac{1}{z}\right)=\mathrm{a}+\rho(\\{0\\})z+\int_{\check{A}}\frac{zx-1}{z+x}\check{\rho}(dx)\in\mathcal{N}(\check{A}).$

Note that in (8), we need to know $F$ in some strips of the complex plane to
retrieve a measure $\rho$. A natural question is:

What can be said about a measure $\rho$ if we only have values $F(iw)$, for
$w\neq 0,$ and we
don’t know if it is a restriction of an analytic function to the imaginary
axis?

The answer was given by Jankowski and Jurek (jan, , Theorem 1), where there is
an inversion procedure, which allows to identify the measure $\rho$, or more
precisely its Fourier transform $\mathfrak{F}[\rho,s],\;s\in\mathds{R}$, and
which is justified as follows: let us, for an index $X$ (where $X$ can be a
random variable, or a stochastic process or a measure), define the function
$F_{{\scriptscriptstyle X}}$ on the imaginary axis
$i(\mathds{R}\setminus{\\{0\\}})$ by

$F_{{\scriptscriptstyle X}}(iw):=\mathrm{a}_{{\scriptscriptstyle
X}}+i\,\mathrm{b}_{{\scriptscriptstyle
X}}\,w+\int_{\mathds{R}}\frac{iwx-1}{iw+x}\rho_{{\scriptscriptstyle
X}}(dx),\quad w\neq 0,$ (10)

where $\mathrm{a}_{{\scriptscriptstyle X}}\in\mathds{R}$,
$\mathrm{b}_{{\scriptscriptstyle X}}\geq 0$ and $\rho_{{\scriptscriptstyle
X}}$ is a non-negative, finite Borel measure. Furthermore, if

$\mathfrak{F}[\rho_{{\scriptscriptstyle
X}}\,;s\,]:=\int_{\mathds{R}}e^{isx}\rho_{{\scriptscriptstyle X}}(dx),\quad
s\in\mathds{R},$

is the Fourier transform (respectively the characteristic function) of a
finite measure (respectively probability measure) $\rho_{{\scriptscriptstyle
X}}$, then the _Laplace transform_ of $\mathfrak{F}[\rho_{{\scriptscriptstyle
X}}\,;s\,]$ satisfies the equality

$\displaystyle\mathfrak{L}[\,\mathfrak{F}[\rho_{{\scriptscriptstyle
X}}\,;s\,]\,;\,w]$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}e^{-ws}\,\mathfrak{F}[\rho_{{\scriptscriptstyle
X}}\,;s\,]ds=\int_{\mathds{R}}\frac{1}{w-ix}\,\rho_{{\scriptscriptstyle
X}}(dx),\quad w>0$ (14) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llcr}\frac{1}{w^{2}-1}\,\big{[}iF_{{\scriptscriptstyle
X}}(iw)-i\Re\big{(}F_{{\scriptscriptstyle
X}}(i)\big{)}+w\,\Im\big{(}F_{{\scriptscriptstyle
X}}(i)\big{)}\big{]}&\mbox{if}&w\neq 1\\\ &&\\\
\displaystyle\int_{\mathds{R}}\frac{1+ix}{1+x^{2}}\,\rho_{{\scriptscriptstyle
X}}(dx)&\mbox{if}&w=1.\end{array}\right.$

Motivated by the special structure of the Nevanlinna-Pick functions appearing
in (14), we focus in Section 2 on their analytical aspect and provide
additional clarification to their relation to classical and free probability
context. An appendix is included to explain properties of some used special
functions and Section 5 is devoted to the proofs.

Our main results are obtained in sections 3 and 4, where we study two aspects
of Nevanlinna-Pick functions:

  * •

The first studied aspect, is mainly probabilistic. In Section 3 we generalize
the connection described by Schilling, Song & Vondraček (SSV, , Theorem 6.9),
between the subclass of nonnegative functions in $\mathcal{N}$ and the one of
Bernstein functions associated to subordinators. We provide an connection of
the same nature between a subclass of Nevanlinna-Pick functions and spectrally
negative Lévy processes. This connection is also of same nature than the one
obtained right after (33) below when replacing Bernstein functions by Lévy-
Laplace exponents. We also provide a property of temporal complete
monotonicity for subordinators as Bertoin and Yor by  did for self-similar
Markov processes obtained via the Lamperti transform, cf. (37) below. During
several discussions of the first author with Loïc Chaumont, a recurrent
question was:

Let $\phi$ be the Bernstein function of some subordinator $\xi$ such that is
impossible to check if $\Psi=\phi^{-1}$ is a Lévy-Laplace exponent by standard
calculations. Are there any other clue indicating whether $\xi$ is the first
passage time of some spectrally negative Lévy process? (15)

Motivated by this question, we provide in Theorem 3.1 a temporal complete
monotonicity property: Any subordinator $\xi={(\xi_{t})}_{t\geq 0}$, such that
its $\Psi$-function satisfies some integrability conditions, satisfies one the
following:

    1. (i)

$t\mapsto\mathbb{E}[\xi_{t}^{-p}]\;$ is a Bernstein function, for some
$\,p\in(-1,0)$;

    2. (ii)

$t\mapsto\mathbb{E}[\xi_{t}^{-p}]\;$ is a completely monotone function, for
some $\,p\in[0,1)$;

    3. (iii)

$t\mapsto t\;\mathbb{E}[\xi_{t}^{-p}]\;$ is a completely monotone function,
for some $\,p\geq 1$.

In Corollary 1, we answer to question (15) by showing that a subordinator
$\xi$ is the inverse times of a spectrally negative Lévy process, if, and only
if,

$t\mapsto t\,\mathbb{E}[\xi_{t}^{-p}]\;\;\mbox{is a {\it Stieltjes transform},
for some $\,p\geq 1$}.$

The latter is also equivalent to:

$t\mapsto t^{2}\,\mathbb{E}[\xi_{t}^{-p}]\;\;\mbox{is a {\it complete
Bernstein function}, for some $\,p\geq 1$}.$

Of course, one can argue that the calculus of the moments
$\mathbb{E}[\xi_{t}^{-p}],\,t>0,\,p>-1,$ are also not always feasible, but,
when they are, we immediately have a conclusion, see Example 2 and especially
Example 3 . To the best of our knowledge, the temporal properties that we
obtain are new.

  * •

The second aspect, studied in Section 4, is related to the free-probability
context, and consists in the application of the inversion formula (35). In
Propositions 3, 4, 5 and Corollary 2, below, we show that for some related
index $X$, the functions

$w\mapsto\frac{iF_{{\scriptscriptstyle X}}(iw)}{w^{2}-1},\quad w>1,$

in (14), are indeed, Laplace transforms of some functions or measures that, in
principle, enables us to identify the corresponding measure
$\rho_{{\scriptscriptstyle X}}$. Let $X=C,\,S$ and $T$ be random variables
defined by their hyperbolic characteristic functions

$\mathfrak{F}_{{\scriptscriptstyle
X}}(s):=\mathbb{E}[e^{isX}]=\int_{\mathds{R}}e^{isx}\,\mathbb{P}(X\in
dx),\quad s\in\mathds{R},$ $\mathfrak{F}_{{\scriptscriptstyle
C}}(s):=\frac{1}{\cosh(s)},\quad\mathfrak{F}_{{\scriptscriptstyle
S}}(s):=\frac{s}{\sinh(s)},\quad\mathfrak{F}_{{\scriptscriptstyle
T}}(s):=\frac{\tanh(s)}{s}.$

and

$\mbox{$\tilde{X}$ be the {\it free-analog} of the of $X,$ obtained by the
procedure {\bf(P4)} in Section \ref{free}}.$

We emphasize that in our approach to free-probability theory, we only use
purely imaginary numbers $\,z=iw,\,w>1$; cf. (35) and Jurek (j2, , Corollaries
3, 4 and 5) and we recall that the hyperbolic characteristic functions were
studied:

    1. (i)

from an infinite divisibility point of view, in Pitman and Yor Pitman ;

    2. (ii)

from a self-decomposability point of view in Jurek j1 , as infinite series of
independent exponentially distributed variables;

    3. (iii)

from stochastic representations of their background driving Lévy processes
(BDLP) in Jurek-Vervaat (jv, , Theorem 3.2). The last can be done since all
hyperbolic characteristic functions are self-decomposable ones and therefore,
they admit a representation by random integrals

$X=\int_{0}^{\infty}e^{-t}\,dY_{t},\quad\mbox{where $Y:=Y_{{\scriptscriptstyle
X}}$ is the BDLP},$ (16)

see also Jurek and Mason (jm, , Chapter 3, Theorem 3.6.8).

##  2 Various interpretations of Nevanlinna-Pick functions

We first list several interesting properties of Nevanlinna-Pick functions.

  1. (P1)

Every Nevanlinna-Pick function $F$ in (5) is represented by

$F(z)=\mathrm{a}+\mathrm{b}z+\int_{\mathds{R}}f(x,z)\,\rho(dx),\quad
z\in\mathds{C}\setminus\mathds{R},$

where the functions

$z\mapsto f(x,z)=\frac{xz-1}{z+x},\quad
x\in\mathds{R},\;z\in\overline{\mathds{C}}\backslash\\{-x\\}$ (17)

are the it Möbius mappings, which are conformal (i.e analytic with non-null
derivative) on any region contained in
$\overline{\mathds{C}}\backslash\\{-x\\}$ and then, preserve the angles in
this region. Roughly speaking, $f$ preserve the angles means that for any two
rays $L$ and $L^{\prime}$, starting at a point $z_{0}$, the angle which their
images $f(L)$ and $f(L^{\prime})$ make at $z_{0}$ is the same as that made by
$L$ and $L^{\prime}$, in size as well as in orientation. These mappings enjoy
the following properties:

$\displaystyle(a)$ $\displaystyle f(x,i)=i;$ $\displaystyle(b)$
$\displaystyle\Im\big{(}f(x,z)\big{)}=\frac{1+x^{2}}{|x+z|^{2}}\Im(z)\;\,\Longrightarrow\;\,f(x,\mathds{H}^{+})\subset\mathds{H}^{+}\;\;\mbox{and}\;\;f(x,\mathds{H}^{-})\subset\mathds{H}^{-}\,;$
$\displaystyle(c)$ $\displaystyle f(x,\overline{z})=\overline{f(x,z)};$
$\displaystyle(d)$ $\displaystyle f(x,-z)=-f(-x,z);$ $\displaystyle(e)$
$\displaystyle f\left(\frac{1}{x},\frac{1}{z}\right)=-f(x,z),\quad x,\,z\neq
0.$

  2. (P2)

For every $\alpha>0$ there exists $N_{\alpha}$ such that $F$ has a left
inverse $F^{-1}$ defined on the region $\Gamma_{\alpha,N_{\alpha}}$ given by
(4). cf. (bv, , Proposition 5.4 and Corollary 5.5).

  3. (P3)

Any Cauchy transforms, i.e., a function of the form

$G_{\mu}(z)=\mathrm{c}+\int_{\mathds{R}}\frac{1}{z+x}\mu(dx),\quad
z\in\mathds{C}{\footnotesize\setminus}\mathds{R},$ (18)

where $\mathrm{c}$ is a real number and $\mu(dx)=(1+x^{2})\rho(dx)$ is a
finite measure, satisfies $-G\in\mathcal{N}(\mathds{R})$. To see that, it is
enough to take $\mathrm{b}=0$ in (6) and to replace
$\mathrm{a}+\int_{\mathds{R}}x\,\rho(dx)$ by $-\mathrm{c}$. In other terms,
$G$ swaps the upper and lower complex half planes. In this case, $z\mapsto
G_{\mu}(1/z)\in\mathcal{N}(\mathds{R})$. To see this, use property (c) of the
function $f$ in (17) or the half-plane swapping property, which also yields
$H_{\mu}=1/G_{\mu}\in\mathcal{N}(\mathds{R})$. Now, assume $\mathrm{c}=0$ and
observe that

$\mathrm{d}:=\mu(\mathds{R})=\lim_{|z|\to\infty,\,z\in\Gamma_{\alpha,N}}z\,G_{\mu}(z)>0,\quad\mbox{for
every}\;\alpha,\;N>0.$ (19)

The finite measure $\mu$ is called (additive)-free infinitely divisible if,
and only if, the function $H_{\mu}=1/G_{\mu}$ is such that its inverse
$H_{\mu}^{-1}$ (which, by (P2) in Section (2), always exists) satisfies

$z\mapsto
F_{\mu}(z):=\mathrm{d}\,z-H_{\mu}^{-1}(z)\in\mathcal{N}(\mathds{R}).$ (20)

The function $V_{\mu}(z)=-F_{\mu}$ is commonly called the Voiculescu transform
of $\mu$, cf. (9) and Bercovici and Voiculescu bv . The free probability
context corresponds to probability measures $\mu$ and the parameter
$\mathrm{d}$ in (18) is set to be equal 1.

  4. (P4)

Let $\Psi$ be a Lévy-Laplace exponent, i.e., a function represented by

$\Psi(\lambda)=\alpha+\beta\lambda+\frac{\gamma^{2}}{2}\lambda^{2}+\int_{\mathds{R}\setminus\\{0\\}}\left(e^{-\lambda
x}-1+\lambda\frac{x}{1+x^{2}}\right)\nu(dx),\quad\lambda\in
i\,\cdot\,\mathds{R},$ (21)

where $\alpha,\,\gamma\geq 0$, $\beta\in\mathds{R}$, and the so-called Lévy
measure $\nu$ is supported by $\mathds{R}\setminus\\{0\\}$ and such that

$\int_{\mathds{R}}\left(x^{2}\wedge
1\right)\nu(dx)<\infty\Longleftrightarrow\int_{\mathds{R}}\frac{x^{2}}{1+x^{2}}\nu(dx)<\infty.$
(22)

As we did for the Nevanlinna class, we denote

$\mathcal{LE}(A):=\big{\\{}\mbox{Laplace
exponents}\,\;\Psi\;\;\mbox{represented by \eqref{EL},
s.t.}\;support(\nu)\subset A\big{\\}},\quad A\subset\mathds{R}.$ (23)

For the set $\mathcal{LE}(0,\infty)$, we have the following stochastic
interpretation: there is a bijection between the class of (killed) Lévy
process, i.e. a process $Z={(Z_{t})}_{t\geq 0},\;Z_{0}=0$, with stationary and
independent increments, and the class of Lévy-Laplace exponent
$\Psi_{{\scriptscriptstyle X}}$, represented by (21), via the celebrated Lévy-
Khintchine formula, i.e. the Laplace representation: if $X:=Z_{1}$, then

$\mathbb{E}[e^{\lambda Z_{t}}]=e^{t\Psi_{{\scriptscriptstyle
X}}(\lambda)},\qquad\mbox{for}\;\,t\geq 0,\;\;\lambda\in i.\mathds{R},$

Actually, the distributions of $Z_{t},\,t>0$, are entirely determined by the
one of the infinitely divisible random variable $X=Z_{1}$. In (21), the usage
is to call the quantity $\alpha_{{\scriptscriptstyle X}}$ the killing rate,
$\beta_{{\scriptscriptstyle X}}$ the drift term and
$\gamma_{{\scriptscriptstyle X}}$ the Brownian coefficient. To every function
Nevanlinna-Pick function $F$, represented by (10), one can associate a free-
infinitely divisible random variable $\tilde{X}$ that we call free-analog of
$X$ by these means:

     * •

Pick a Voiculescu transform of some free-infinitely divisible random variable
$\tilde{X}$ whose distribution is the probability measure $\mu$ in obtained by
(20).

     * •

Barndorff-Nielsen and Thorbjørnsen (BT, , Theorem 4.1) and Jurek (j2, ,
Theorem 1), clarified, in a more involved way, the so-called Bergovici-Pata
bijection from the class of Lévy processes to the one of free-infinitely
divisible random variables, cf. Bercovici, Pata & Biane BP . More precisely,
there exists an infinitely divisible random variable $X$ with Lévy-Laplace
exponent $\Psi_{{\scriptscriptstyle X}}$, represented by (21), such that the
function

$F_{{\scriptscriptstyle\tilde{X}}}(iw)=i\,w^{2}\int_{0}^{\infty}e^{-wu}\,\Psi_{{\scriptscriptstyle
X}}(-iu)\,du,\quad w>0.$ (24)

     * •

Observe that the Laplace transform of $\Psi_{{\scriptscriptstyle X}}(-iu)$
always exists since it is a continuous function which satisfies
$\lim_{|u|\to\infty}\Psi_{{\scriptscriptstyle X}}(-iu)/u^{2}=-\gamma^{2}/2$.
The latter is justified by the control of the kernel: for $\Re(\lambda)\geq
0,\;x\neq 0$, we have

$\displaystyle e^{-\lambda x}-1+\lambda\frac{x}{1+x^{2}}$ $\displaystyle=$
$\displaystyle(e^{-\lambda x}-1+\lambda x)\,{\rm 1\leavevmode\nobreak\
l}_{|x|\leq 1}+(e^{-\lambda x}-1)\,{\rm 1\leavevmode\nobreak\ l}_{|x|\geq
1}+\lambda\left(\frac{x^{2}}{1+x^{2}}\,{\rm 1\leavevmode\nobreak\ l}_{|x|\leq
1}-\frac{x}{1+x^{2}}\,{\rm 1\leavevmode\nobreak\ l}_{|x|\geq 1}\right)$ (25)

and for $\lambda\geq 0$ or $\lambda=iu,\,u\in\mathds{R}$, we have

$\displaystyle\left|e^{-\lambda x}-1+\lambda\frac{x}{1+x^{2}}\right|$
$\displaystyle\leq$ $\displaystyle\frac{|\lambda|^{2}\,x^{2}}{2}\,{\rm
1\leavevmode\nobreak\ l}_{|x|\leq 1}+2\,{\rm 1\leavevmode\nobreak\ l}_{|x|\geq
1}+|\lambda|\,\left(\frac{x^{2}}{1+x^{2}}\,{\rm 1\leavevmode\nobreak\
l}_{|x|\leq 1}+\frac{x}{1+x^{2}}\,{\rm 1\leavevmode\nobreak\ l}_{|x|\geq
1}\right)\leq C(\lambda)\,(x^{2}\wedge 1),$

where $C(\lambda):=3+|\lambda|+\frac{|\lambda|^{2}}{2}$. The latter insures
that $\Psi_{{\scriptscriptstyle X}}$ is well defined due to the condition (22)
on its Lévy measure. Thus, the triplet of characteristics of
$F_{{\scriptscriptstyle\tilde{X}}}$ are given by

$\mathrm{a}_{{{\scriptscriptstyle\tilde{X}}}}=\beta_{{\scriptscriptstyle
X}},\quad\mathrm{b}_{{{\scriptscriptstyle\tilde{X}}}}=\alpha_{{\scriptscriptstyle
X}}\quad\mbox{and}\quad\rho_{{{\scriptscriptstyle\tilde{X}}}}(dx)=\gamma_{{\scriptscriptstyle
X}}^{2}\delta_{0}(dx)+\frac{x^{2}}{1+x^{2}}\nu_{{\scriptscriptstyle X}}(dx),$
(26)

$\rho_{{{\scriptscriptstyle\tilde{X}}}}$ is a finite measure and

the operator $\Delta:\Psi\mapsto F$ defined by (24), induces a bijection from
$\mathcal{LE}(0,\infty)$ to $\mathcal{N}(\mathds{R})$, or equivalently,
from the class of Lévy processes to the one of free-infinitely divisible
random variables.

###### Remark 1

Assume $X:\stackrel{{\scriptstyle d}}{{=}}X_{1}+X_{2}$ where $X_{1}$ and
$X_{2}$ are independent infinitely divisible random variables. On the level of
the Lévy measures, we have $\nu_{{\scriptscriptstyle
X}}=\nu_{X_{1}}+\nu_{X_{2}}$ and (26) gives
$\rho_{{\scriptscriptstyle\tilde{X}}}=\rho_{{\scriptscriptstyle\tilde{X}}_{1}}+\rho_{{\scriptscriptstyle\tilde{X}}_{2}}$
on the level of the Nevanlinna measures.

  5. (P5)

If $F$ has a continuous extension with $F:(0,\infty)\to\mathds{R}$, then
$F\in\mathcal{N}[0,\infty)$. Further, if

$F(0+)=\mathrm{a}-\int_{(0,1]}\frac{1}{x}\rho(dx)$

is finite, then $F\in\mathcal{N}(0,\infty)$, cf. (SSV, , Theorem 6.9).

  6. (P6)

The class $\mathcal{S}$ of Stieltjes transforms is formed by functions with a
represented similar to (18): $f\in\mathcal{S}$, if

$f(\lambda)=\mathrm{d}+\frac{\mathrm{q}}{z}+\,\int_{(0,\infty)}\frac{1}{z+u}\;\Delta(du),\quad
z\in\mathds{C}_{+},$ (27)

where $\mathrm{d},\mathrm{q}\geq 0$ are constants and
$\int_{(0,\infty)}(1+u)^{-1}\;\Delta(du)<\infty$, cf. (SSV, , Definition 2.1).
The class $\mathcal{S}$ is simply seen as the class of double iterated Laplace
transforms, due to the representation

$\frac{1}{1+z}\int_{0}^{\infty}e^{-zx}\,e^{-x}\,dx,\quad z\geq 0$

Using the the exponential integral function ${\rm Ei}$ defined by (85) in the
Appendix, an example is given by the function ${\mathfrak{Ei}}(z):-e^{z}\,{\rm
Ei}(-z)$ represented by

${\mathfrak{Ei}}(z):=\int_{0}^{\infty}\frac{e^{-x}}{z+x}\,dx,\quad z>0.$

The function

$\mathfrak{S}(z)=\frac{\log
z}{z-1}=\int_{0}^{\infty}\frac{1}{z+1}\,\frac{1}{x+1}\,dx,\quad z>0,$

is another important example of a Stieltjes function and, actually, it is a
double iterated Stieltjes function. As for the class $\mathcal{S}$, it appears
as the kernel of the class s $\mathcal{S}_{2}$, of double iterated Stieltjes
transforms of functions, whose several properties were provided by Yakubovich
and Martins YM . This class could be extended to iterated Stieltjes transforms
of measures

$\mathcal{S}_{2}:=\left\\{f(z)=\int_{(0,\infty)}\mathfrak{S}(zt)\;\Delta(dt),\quad
z>0\right\\}.$

We believe that it might be interesting to study the properties of the class
of iterated Laplace transform of any order, and we hope that this will be the
scope of a future work.

By Theorem (SSV, , Theorem 6.2, Proposition 7.1, Theorem 7.3), the class
$\mathcal{CBF}$ of complete Bernstein functions is formed by those functions
$f$ of one the following forms

$f(z)=z\,g(z)=h\left(\frac{1}{z}\right)=\frac{1}{k(z)},\quad\mbox{where}\quad
g,\,h,\,k\in\mathcal{S}.$

Notice that $\mathcal{CBF}$ is contained into the class $\mathcal{BF}$ of
Bernstein functions formed by functions $\phi$ of the form:

$\phi(z)=\mathrm{q}+\mathrm{d}z+\int_{(0,\infty)}(1-e^{-zx})\,\Pi(dx),\quad
z\geq 0$ (28)

where $\mathrm{q}\\\ geq0$ is the so-called killing rate,
$\mathrm{d}\in\mathds{R}$ is the drift and $\Pi$ is the Lévy measure of
$\phi$, i.e. a positive measure on $(0,\infty)$ which satisfies
$\int_{(0,\infty)}\left(x\wedge 1\right)\Pi(dx)<\infty$ and that
$\phi^{\prime}(z)$ is a completely monotone function. We recall that a
function $f$ is completely monotone function on $(0,\infty)$ if $f$ in
infinitely often differentiable and $(-1)^{n}f^{(n)}\geq 0$, for all
$n=0,1,2,\cdotp$. equivalently $f$ is the Laplace transform of some measure,
represented by

$f(\lambda)=\int_{[0,\infty)}e^{-\lambda
x}\mu(dx)=\lambda\int_{0}^{\infty}e^{-\lambda x}\mu([0,x))\,dx.$ (29)

Cf. (SSV, , Chapter 1). Complete Bernstein function were defined in (SSV, , p.
69) as those Bernstein function such that their Lévy measure $\Pi$ have the
form

$\Pi(dx)=m(x)\,dx,\;>0,\quad\mbox{where $m$ is a completely monotone
function},$ (30)

i.e. $m(x)=\mathfrak{L}[\sigma\,;\,x]$. Thus, every $f\in\mathcal{CBF}$ takes
the $\mathcal{N}(0,\infty)$ form:

$f(\lambda)=\mathrm{q}+\mathrm{d}\lambda+\int_{(0,\infty)}\frac{\lambda}{\lambda+x}\sigma(dx),\quad\lambda\geq
0,$ (31)

where $\mathrm{q},\,\mathrm{d}\geq 0$ and $\sigma$ is some measure on
$(0,\infty)$ that integrates $1/(1+x)$. From the proof of (SSV, , Theorem
6.9), one can extract that the Nevanlinna-characteristics of $f$ are

$\mathrm{a}_{f}=\mathrm{q}+\int_{(0,\infty)}\frac{\sigma(dx)}{1+x^{2}},\quad\mathrm{b}_{f}=\mathrm{d}\quad\mbox{and}\quad\rho_{f}(dx)=\frac{x}{1+x^{2}}\sigma(dx).$
(32)

By (SSV, , Theorem 6.2), the extension of $f$ is also represented by a
Nevanlinna representation analogous of (24):

$f(z)=z^{2}\int_{0}^{\infty}e^{-zx}\,\phi(x)\,dx=z^{2}\,\mathfrak{L}[\phi,z],\quad
z\in\mathds{C}^{+},$ (33)

where $\phi$ is a some Bernstein function. The latter induces that

the operator $\phi\mapsto f$ given by (33), is a bijection from $\mathcal{BF}$
to $\mathcal{CBF}$, or equivalently, from the class
of subordinators to the one of complete subordinators.

By (SSV, , Theorem 6.9), we extract that actually, the analytic extension on
$\mathds{C}\setminus\mathds{R}_{+}$ of $\mathcal{CBF}$-functions corresponds
to those functions $F$ in $\mathcal{N}(0,\infty)$ which are nonnegative-valued
on $(0,\infty)$. A non trivial example of a $\mathcal{CBF}$-function is given
by the following: let ${\rm Li}_{s}$ be the polylogarithm functions given by
(90) and define

${\mathfrak{Li}}_{s}(z):=-{\rm Li}_{s}(-z),\quad z\geq 0,\;\,\;s>0.$

By (92), ${\mathfrak{Li}}_{s}$ meets the $\mathcal{CBF}$-form (31):

${\mathfrak{Li}}_{s}(z)=\frac{1}{\Gamma(s)}\int_{1}^{\infty}\frac{z}{z+x}\,\log^{s-1}(x)\,\frac{dx}{x},\quad
z\geq 0.$

If furthermore $s\geq 1$, ${\mathfrak{Li}}_{s}$ is in the class
$\mathcal{TBF}$ of Thorin Bernstein functions, the subclass of
$\mathcal{CBF}$-functions, such that the $m$-function in (30)

$\Pi(dx)=m(x)\,dx,\;x>0,\quad\mbox{where $x\,m(x)$ is a completely monotone
function},$ (34)

cf. (SSV, , Theorem 8.2 (v)). In general, ${\mathfrak{Li}}_{s},\,s>0,$ meets
the Nevanlinna $\mathcal{N}[1,\infty)$-form (5) on
$\mathds{C}{\footnotesize\setminus}[-1,\infty)$ and its characteristics are
given by (32):

$\mathrm{a}=\frac{1}{\Gamma(s)}\int_{1}^{\infty}\frac{\log^{s-1}(x)}{x(1+x^{2})}\,dx,\quad\mathrm{b}=0\quad\mbox{and}\quad\rho(dx)=\frac{1}{\Gamma(s)}\,\frac{\log^{s-1}(x)}{1+x^{2}}\,dx,\;\;x\geq
1.$

By formula (Lewin, , (1.110) p. 27), observe that ${\mathfrak{Li}}_{2}$ and
${\mathfrak{Ei}}$ are linked by a formula of type (33):

$z\mapsto
z^{2}\int_{0}^{\infty}e^{-zx}\,{\mathfrak{Li}}_{2}(x)\,dx=z\int_{z}^{\infty}\frac{{\mathfrak{Ei}}(y)}{y}\,dy\,\in\mathcal{CBF}.$

  7. (P7)

The class $\mathcal{CBF}_{\\!\\!e}$ of extended complete Bernstein functions
is formed by those functions $f$ of the form $f=g-h$, where
$g\in\mathcal{CBF}$ and $h\in\mathcal{S}$. Actually, the analytic extension on
$\mathds{H}^{+}$ of $\mathcal{CBF}_{\\!\\!e}$-functions corresponds to those
functions $F$ in $\mathcal{N}[0,\infty)$ which are real-valued on
$(0,\infty)$. See (SSV, , Proposition 6.12) and Remark 2 below.

After these preliminaries, we are able to state our first result which is a
generalisation of the inversion procedure (14) and of (SSV, , Corollary 6.13)
stated for $\mathcal{CBF}_{\\!\\!e}$-functions.

###### Theorem 2.1

Let $A\subset\mathds{R}$ and $F:\mathds{H}^{+}\to\mathds{H}^{+}$ be a an
analytic function and recall $\mathfrak{L}$ and $\mathfrak{F}$ stand of the
Laplace and Fourier transforms. Then, the following statements are equivalent:

  1. 1)

$F\in\mathcal{N}(A)$;

  2. 2)

$\displaystyle
G(z):=\frac{1}{1+z^{2}}\,\left[\Re\big{(}F(i)\big{)}+\Im\big{(}F(i)\big{)}\,z-F(z)\right],\;z\in\mathds{H}^{+},\;$
is the Cauchy transform of a finite measure on $A$.

  3. 3)

The function $\displaystyle H(w):=\frac{iF(iw)}{w^{2}-1},\;\,w>1$, is of the
form

$H(w)=\mathfrak{L}\left[\mathfrak{F}[\rho\,;\,s]+i\Re\big{(}F(i)\big{)}\sinh(s)-\Im\big{(}F(i)\big{)}\cosh(s);w\right].$
(35)

for some finite measure $\rho$ on $A$.

In particular $F\in\mathcal{N}(\mathds{R}_{+}),\;$ if, and only if the
function $G$ has continuous extension $\;G:(0,\infty)\to\mathds{R}_{+}$ which
is the Stieltjes transform of some finite measure on $\mathds{R}_{+}$.

##  3 The $\Theta$-transform, Nevanlinna-Pick functions and Lévy processes

A (possibly killed) subordinator $\xi={(\xi_{t})}_{t\geq 0}$ started from 0 is
an increasing Lévy process whose distribution are also obtained via a Lévy-
Khintchine formula

$\mathbb{E}[e^{-\lambda\xi_{t}}]=\int_{[0,\infty)}e^{-\lambda
x}\,\mathbb{P}(\xi_{t}\in dx)=e^{-t\,\phi(\lambda)},\quad\mbox{
for}\;t,\;\lambda\geq 0.$ (36)

where $\phi$ is Bernstein function, i.e. a function represented by (28). See
the monograph of Bertoin bertoin  for background on subordinators and the book
of Schilling, Song & Vondraček SSV  SSV  for Bernstein functions. Lamperti’s
transformation uses the implicit time-change $\tau_{t},\;t\geq 0$, defined by
the identity

$t=\int_{0}^{\tau_{t}}e^{\xi_{s}}\,ds,$

where $\xi$ is a general subordinator. The process $E={(E_{t})}_{t\geq
0}:={(e^{\xi_{\tau_{t}}})}_{t\geq 0}$ is a strong Markov process started from
1 which enjoys the scaling property: if $\mathbb{P}_{x},\;x>0$, is the
distribution of the process $x\,E_{t/x},\;t\geq 0$, then $\mathbb{P}_{x}$
coincides with the law of the process $E$ started from $x$ (that is, when the
subordinator $\xi$ is replaced by $\xi+\log x$ in Lamperti’s transform). In by
, it was shown that for all $p>0$, the function
$t\mapsto\mathbb{E}[E_{t}^{-p}],\;t\geq 0$, is completely monotone with
Laplace transform representation

$\mathbb{E}[E_{t}^{-p}]=\int_{[0,\infty)}e^{-tx}\,\sigma_{p}(dx),$ (37)

where the entire moments of the probability measure $\sigma_{p}$ are given by

$\int_{[0,\infty)}x^{k}\,\sigma_{p}(dx)=\phi(p)\,\phi(p+1)\cdots\phi(p+k-1),\quad
k=1,2\cdots$

A proper subclass of subordinators is formed by first passage times of
spectrally negative Lévy processes $Z={(Z_{t})}_{t\geq 0}$. That means that
$Z$ is a Lévy process with no positive jumps and the distribution of $Z_{t}$
is obtained by a Lévy-Laplace exponent $\Psi$ such that $\mathbb{E}[e^{\lambda
Z_{t}}]=e^{t\Psi(\lambda)}$ and in (21), the Lévy measure $\nu$ of $\Psi$ is
supported by $(0,\infty)$. This entails that $\Psi$ has the continuous
extension on $\mathds{R}_{+}$:

$\Psi(\lambda)=\alpha+\beta\lambda+\frac{\gamma^{2}}{2}\lambda^{2}+\int_{(0,\infty)}\left(e^{-\lambda
x}-1+\lambda\frac{x}{1+x^{2}}\right)\nu(dx),\quad\lambda\geq 0.$ (38)

i.e. $\Psi\in\mathcal{LE}(0,\infty)$. Observe that if in (38), $\nu$ satisfies

$\overline{\beta}:=\beta+\int_{(0,\infty)}\frac{x}{1+x^{2}}\nu(dx)\quad\mbox{is
a finite quantity,}$ (39)

then $\Psi$ satisfies

$\lambda\mapsto\varphi(\lambda):=\alpha+\overline{\beta}\lambda+\frac{\gamma^{2}}{2}\lambda^{2}-\Psi(\lambda)=\int_{(0,\infty)}(1-e^{-\lambda
x})\nu(dx)\in\mathcal{BF}.$ (40)

Conversely, every Bernstein function $\phi$ satisfies

$\lambda\mapsto\alpha+\beta\lambda+\frac{\gamma^{2}}{2}\lambda^{2}-\phi(\lambda)\in\mathcal{LE}(0,\infty),\qquad\mbox{for
all}\;\alpha,\,\gamma\geq 0,\;\beta\in\mathds{R}.$

By the proof of (kyp, , Theorem 3.12 and Section 8.1), under the assumption
that

$Z\;\mbox{is non-killed,
(i.e.}\;\alpha=\Psi(0)=0)\quad\mbox{and}\quad\mathbb{E}[Z_{1}]=\Psi^{\prime}(0+)=\beta-\int_{(0,\infty)}\frac{x^{3}}{1+x^{2}}\,\nu(dx)\geq
0,$ (41)

we have unique (nonnegative) solution $\phi(\mu)$ to the equation
$\Psi(\lambda)=\mu$, given by the right inverse

$\phi(\mu)=\sup\\{\lambda\geq 0\,:\,\Psi(\lambda)=\mu\\}.$ (42)

Necessarily $\phi(0)=0$, and $\phi$ is a Bernstein function represented as in
(28). Actually $\phi$ corresponds to a subordinator $\xi$ which is the first
passage of $Z$ time above a level $s>0$, i.e.

$\xi_{s}=\inf\\{t>0\,:\,Z_{t}>s\\}.$ (43)

Due to (25), observe that for every $\Psi\in\mathcal{LE}(\mathds{R}_{+})$,
there exists $A,\,B,\,C\geq 0$, such that

$\left|\Psi(\lambda)\right|\leq A+B\lambda+C\lambda^{2},\quad\mbox{for all
$\lambda\geq 0$},$

and then the Laplace transform of $\Psi$ is well defined. With this
observation together with (40) and even if the integrability condition (39) is
not satisfied, we still be able to establish a correspondence between the
Nevanlinna class $\mathcal{N}(\mathds{R}_{+})$ with the class
$\mathcal{LE}(\mathds{R}_{+})$ given by (23) as obtained the one obtained in
(33):

######  Proposition 1 (The $\Theta$-transform)

To every pair
$(\mathrm{b},\Psi)\in\mathds{R}_{+}\times\mathcal{LE}(0,\infty)$, is
associated a (unique) Nevanlinna $\mathcal{N}(\mathds{R}_{+})$-function
obtained by the operator $\Theta$ given, for all $z\in\mathds{C}^{+}$, by

$\Theta(\mathrm{b},\Psi)(z)=F(z):=z^{2}\int_{0}^{\infty}e^{-zu}\,\big{(}\mathrm{b}+\Psi(0)-\Psi(u)\big{)}\,du=\mathrm{a}+\mathrm{b}z+\int_{[0,\infty)}\frac{zx-1}{z+x}\rho(dx),$
(44)

and the characteristics $(\mathrm{a},\rho)$ of $F$ are related to the one’s of
$\,\Psi,\,$ $(\alpha,\,\beta,\,\gamma,\,\nu)$, by

$\mathrm{a}=-\beta,\quad\mbox{and}\quad\rho(dx)=\gamma^{2}\,\delta_{0}(dx)+\frac{x^{2}}{1+x^{2}}\nu(dx).$

Furthermore, the Nevanlinna class of functions

$\mathcal{N}^{\star}(\mathds{R}_{+}):=\left\\{F\in\mathcal{N}(\mathds{R}_{+})\;\;\mbox{s.t.}\;\;\mathrm{b}=\lim_{|z|\to\infty,\;z\in\mathds{C}^{+}}\frac{F(z)}{z}=0\right\\},$

is obtained as the image of $\\{0\\}\times\mathcal{LE}(0,\infty)$ by the
operator $\Theta$.

###### Remark 2

As in (P7) in Section 2, one can see that the class

$\mathcal{CBF}_{\\!\\!e}^{\star}:=\\{f\in\mathcal{CBF}_{\\!\\!e},\;\mbox{such
that}\;\lim_{\lambda\to\infty}f(\lambda)/\lambda=0\\}$

is such that the extension on $\mathds{H}^{+}$ of all its members $f$
coincides with the class $\mathcal{N}^{\star}(\mathds{R}_{+})$ given in last
proposition.

######  Example 1 (The $\Theta$-transform for spectrally negative stable
processes)

Let $\vartheta\in(1,2]$. The function
$\Psi_{\vartheta}(\lambda):=\lambda^{\vartheta},\;\lambda\geq 0$, multiplied
by positive constants, corresponds to the Lévy-Laplace exponents of spectrally
negative stable processes, see (kyp, , Section 6.5.3). Observe that $\Psi_{2}$
corresponds to the Gaussian distribution and that $\Theta(0,y\mapsto
u^{2})(z)=-2/z^{2}$. For $\vartheta\in(1,2)$, let

$c_{\vartheta}=\frac{\vartheta(\vartheta-1)}{\Gamma(2-\vartheta)}\quad\mbox{and}\quad\rho_{\vartheta}(x)=\frac{c_{\vartheta}\,x^{1-\vartheta}}{1+x^{2}},\;x>0.$

Using (grad, , Formula FI II 718 p. 319) and Euler’s reflection formula, we
easily get that

$\beta_{\vartheta}\>=c_{\vartheta}\int_{0}^{\infty}\frac{x^{2-\vartheta}}{1+x^{2}}\,dx=\frac{\vartheta}{\Gamma(2-\vartheta)}\Gamma\left(\frac{\vartheta+1}{2}\right)\Gamma\left(\frac{3-\vartheta}{2}\right).$

The function $\Psi_{\vartheta}$ is represented by

$\Psi_{\vartheta}(\lambda)=\int_{0}^{\infty}\left(e^{-\lambda x}-1+\lambda
x\right)\frac{c_{\vartheta}}{x^{\vartheta+1}}\,dx=\beta_{\vartheta}\lambda+\int_{0}^{\infty}\left(e^{-\lambda
x}-1+\lambda\frac{x}{1+x^{2}}\right)\frac{c_{\vartheta}}{x^{\vartheta+1}}\,dx.$

The Nevanlinna-Pick function
$F_{\vartheta}(z)=-\Gamma(\vartheta+1)\,z^{1-\vartheta},\;z\in\mathds{C}^{+}$,
is the $\mathcal{CBF}_{\\!\\!e}$-function obtained by the operator $\Theta$ in
(44). Actually $-F_{\vartheta}$ is a Stieltjes function in the sense of (P6)
in Section 2 and without effort, we get the Nevanlinna representation

$F_{\vartheta}(z)=-\beta_{\vartheta}+\int_{0}^{\infty}\frac{zx-1}{z+x}\,\rho_{\vartheta}(x)\,dx,\quad
z\in\mathds{C}^{+}.$

Lemma 1 in Subsection 5.1 below, is the key result for the following temporal
completely monotone property for subordinators.

######  Proposition 2 (Temporal completely monotonicity property for
subordinators)

Let $\phi:\mathds{R}_{+}\to\mathds{R}_{+}$ be a strictly increasing function
such that $\phi(0)=0$ and $\phi(\infty)=\infty$. Let $\Psi$ be its inverse
function and e denotes a r.v. with standard exponential distribution. Then the
following statements are equivalent.

  1. 1)

$\phi$ is a Bernstein function;

  2. 2)

There exists a subordinator $\xi={(\xi_{t})}_{t\geq 0}$ satisfying the
identities in law

$\frac{\mbox{\bf e}}{\xi_{t}}\stackrel{{\scriptstyle
d}}{{=}}\Psi\left(\frac{\mbox{\bf e}}{t}\right),\quad\forall t>0,\qquad\mbox{
where $\mbox{\bf e}$ in the l.h.s. is assumed to be independent of
$\xi_{t}$};$ (45)

  3. 3)

There exists a family of positive r.v.’s $\eta={(\eta_{t})}_{t\geq 0}$ such
that (45) holds true with $\xi$ replaced by $\eta$;

  4. 4)

There exists a subordinator $\xi={(\xi_{t})}_{t\geq 0}$, such that the
representation

$\mathbb{E}[\xi_{t}^{-p}]=\frac{t}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\,\Psi^{p}(x)\,dx$
(46)

holds true for every $t>0$ and $p>-1$, such that
$\mathfrak{L}[\Psi^{p}\,;\,t]<\infty$.

Under the latter, $\phi$ is necessarily the Bernstein function of $\xi$.

###### Remark 3

Observe that the temporal completely monotone property in (46) involves the
Laplace representation in the right hand side term similar to the one obtained
in (37) with $p=1$. Also observe that $\phi(\lambda)=\lambda$ is associated to
the trivial subordinator $\xi_{t}=t$ , then replacing by
$\Psi(\lambda)=\lambda$ in (46), we retrieve the integral representation of
the Gamma function (84).

######  Example 2 (Moments of positive stable distributions)

If
$\xi^{({\scriptscriptstyle\alpha})}={(\xi^{({\scriptscriptstyle\alpha})}_{t})}_{t\geq
0}$ is a standard $\alpha$-stable subordinator, i.e., with Bernstein function
$\phi_{\alpha}=\lambda^{\alpha}$, $0<\alpha<1$, then (45) gives the following
well known results, valid for all $t>0,\;p>-\alpha$: by (45) , we have

$\frac{\mbox{\bf
e}}{\xi^{({\scriptscriptstyle\alpha})}_{t}}\stackrel{{\scriptstyle
d}}{{=}}\left(\frac{\mbox{\bf
e}}{t}\right)^{\frac{1}{\alpha}}\Longrightarrow\frac{\mbox{\bf
e}}{\xi^{({\scriptscriptstyle\alpha})}_{1}}\stackrel{{\scriptstyle
d}}{{=}}\mbox{\bf
e}^{\frac{1}{\alpha}}\quad\mbox{and}\quad\xi^{({\scriptscriptstyle\alpha})}_{t}\stackrel{{\scriptstyle
d}}{{=}}t^{\frac{1}{\alpha}}\,\xi^{({\scriptscriptstyle\alpha})}_{1}\,\;\mbox{({\it
scaling property for stable processes}),}$ (47)

and by (46) , we have

$\mathbb{E}[(\xi^{({\scriptscriptstyle\alpha})}_{t})^{-p}]=\frac{t}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\,x^{\frac{p}{\alpha}}\,dx=\frac{t}{\Gamma(p+1)}\,\frac{\Gamma(1+\frac{p}{\alpha})}{t^{1+\frac{p}{\alpha}}}=t^{-\frac{p}{\alpha}}\,\frac{\Gamma(1+\frac{p}{\alpha})}{\Gamma(1+p)},$

a formula that in (CY, , Exercise 4.17), whereas the second identity in (47)
is in (CY, , Exercise 4.19). See (jedidi, , Theorem 1) for all moments of all
stable distributions.

###### Remark 4

Proposition 2 shows that if $p>-1$ and $\Psi$ is the inverse of a Bernstein
function $\phi$ associated to the subordinator $\xi$, then the following
assertions are equivalent.

  1. (i)

$t\mapsto\mathbb{E}[\xi_{t}^{-p}]/t$ is completely monotone on $(0,\infty)$

  2. (ii)

$\mathfrak{L}[\Psi^{p}\,;\,]$ is well defined on $(0,\infty)$.

This simple remark is improved as follows:

######  Theorem 3.1 (Temporal completely monotonicity property for
subordinators improved)

Let $p>-1$ and $\phi$ be a non-trivial Bernstein function (i.e. $\phi$ is not
affine) such that $\phi(0)=0,\;\phi(\infty)=\infty$. Let $\Psi$ and
$\xi={(\xi_{t})}_{t\geq 0}$ be respectively, the inverse function and the
subordinator, associated to $\phi$. We have the following results.

  1. 1)

If $p\in(-1,0)$, then $e^{-tx}\,\Psi^{p}(x)$ is integrable near $\infty$ and
the following assertions are equivalent.

    1. (i)

$\Psi^{p}$ is integrable at the neighborhood of $\,0$;

    2. (ii)

$t\mapsto\,\mathbb{E}[\xi_{t}^{-p}]$ is a Bernstein function.

We then have the representation

$\mathbb{E}[\xi_{t}^{-p}]=\frac{1}{\Gamma(p+1)}\int_{0}^{\infty}(1-e^{-tx})\,(-\Psi^{p})^{\prime}(x)\,dx$
(48)

  2. 2)

If $p\geq 0$, then $\Psi^{p}$ is integrable near $\,0$ and the following
assertions are equivalent.

    1. (i)

$x\mapsto e^{-tx}\,\Psi^{p}(x)$ is integrable near $\infty$ for all $t>0$;

    2. (ii)

$t\mapsto\mathbb{E}[\xi_{t}^{-p}]$ is completely monotone on $(0,\infty)$.

We then have the representation

$\mathbb{E}[\xi_{t}^{-p}]=\frac{1}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\,(\Psi^{p})^{\prime}(x)\,dx,\quad
t>0.$ (49)

The function $t\mapsto t\,\mathbb{E}[\xi_{t}^{-p}]$ is Bernstein if, and only
if, $\lambda\mapsto\lambda^{1-p}\phi^{\prime}(\lambda)$ is non-decreasing,
hence $p\in[0,1)$. In this case, we have the representation

$t\mathbb{E}[\xi_{t}^{-p}]=\frac{1}{\Gamma(p+1)}\int_{0}^{\infty}(1-e^{-tx})\;(\Psi^{p})^{\prime\prime}(x)\,dx,\quad
t>0.$ (50)

  3. 3)

If $p\geq 1$, then $\Psi^{p}$ is convex,
$(\Psi^{p})^{\prime}(0+)\in[0,\infty)$, and the following two assertions are
equivalent.

    1. (i)

$x\mapsto\,e^{-tx}\,\Psi^{p}(x)\;$ is integrable near $\infty$ for all $t>0$;

    2. (ii)

$t\mapsto\,t\,\mathbb{E}[\xi_{t}^{-p}]\;$ is completely monotone on
$(0,\infty)$.

In this case, we have the representation

$\displaystyle t\,\mathbb{E}[\xi_{t}^{-p}]$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(p+1)}\left[(\Psi^{p})^{\prime}(0+)+\int_{0}^{\infty}e^{-tx}\,(\Psi^{p})^{\prime\prime}(x)\,dx\right],\quad
t>0,\qquad$ (51)

and the following holds true:

    1. (a)

The function $t\mapsto t^{2}\,\mathbb{E}[\xi_{t}^{-p}]$ is Bernstein if, and
only if, $(\Psi^{p})^{\prime}$ is concave;

    2. (b)

If further $\Psi^{p}\in\mathcal{LE}(0,\infty)$ with quadruple of
characteristics $(0,\,\beta_{p},\,\gamma_{p},\,\nu_{p})$ in its representation
(38), then

$(\Psi^{p})^{\prime}(0+)=\beta_{p}-\int_{(0,\infty)}\frac{x^{3}}{1+x^{2}}\,\nu_{p}(dx)\in[0,\infty),$
(52)

and the function $t\mapsto\,t\,\mathbb{E}[\xi_{t}^{-p}]\;$ has the Stieltjes
transform form (27), i.e.

$t\,\mathbb{E}[\xi_{t}^{-p}]=\frac{1}{\Gamma(p+1)}\left[(\Psi^{p})^{\prime}(0+)+\frac{\gamma_{p}^{2}}{t}+\int_{(0,\infty)}\frac{u^{2}}{t+u}\,\nu_{p}(du)\right],\quad
t>0.$ (53)

###### Remark 5

Under the assumptions of point 3) in Theorem 3.1, we have

$\lim_{t\to\infty}t\,\mathbb{E}[\xi_{t}^{-p}]=\frac{(\Psi^{p})^{\prime}(0+)}{\Gamma(p+1)},$

and under the assumptions of point 4) in Theorem 3.1, we have

$\lim_{t\to\infty}t^{2}\,\mathbb{E}[\xi_{t}^{-p}]-t\frac{(\Psi^{p})^{\prime}(0+)}{\Gamma(p+1)}=(\Psi^{p})^{\prime\prime}(0+)=\gamma_{p}^{2}+\int_{(0,\infty)}u^{2}\,\nu_{p}(du).$

######  Example 3 (Moments of Lambert distributions)

Besides the example of stable subordinators given in Example 2, we propose the
following less trivial one: in Pakes pakes , the principal Lambert function
$\phi_{\scalebox{0.5}{$\mbox{\bf L}$}}$, defined as the unique real-valued
concave increasing solution to the functional equation
$\lambda=\phi_{\scalebox{0.5}{$\mbox{\bf
L}$}}(\lambda)\,e^{\phi_{\scalebox{0.5}{$\mbox{\bf
L}$}}(\lambda)},\;\,\lambda\geq 0,$ is shown to be a Thorin Bernstein, i.e., a
Bernstein function with Lévy measure as in (34). Let
$\xi^{\scalebox{0.5}{$\mbox{\bf L}$}}={(\xi^{\scalebox{0.5}{$\mbox{\bf
L}$}}_{t})}_{t\geq 0}$ be its associated subordinator. Since
$\Psi_{\scalebox{0.5}{$\mbox{\bf L}$}}(x)=x\,e^{x},\;x\geq 0$, then r.v.’s
$\xi^{\scalebox{0.5}{$\mbox{\bf L}$}}_{t}$ satisfy the identity (45):

$\frac{\mbox{\bf e}}{\xi^{\scalebox{0.5}{$\mbox{\bf
L}$}}_{t}}\;\stackrel{{\scriptstyle d}}{{=}}\;\frac{\mbox{\bf
e}}{t}\;e^{\frac{\mbox{\bf e}}{t}}\,\stackrel{{\scriptstyle
d}}{{=}}\;-\frac{1}{t}\;\mbox{\bf u}^{-\frac{1}{t}}\;\log\mbox{\bf
u},\quad\mbox{for all $t>0$},$

where u has the uniform distribution on $(0,1)$, and also (46):

$\mathbb{E}[(\xi^{\scalebox{0.5}{$\mbox{\bf
L}$}}_{t})^{-p}]=\frac{t}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\;(x\;e^{x})^{p}\,dx=\frac{t}{(t-p)^{p+1}},\qquad\mbox{for
all}\;t>0,\;\;t>p>-1.$ (54)

Thus, for all $t>0$, the function

$q\mapsto M_{t}(q):=t\,(t+q)^{q-1},\qquad-t<q<1,$

is a Mellin transform whose domain of definition can be extended to the
interval $(-q,\infty)$, i.e., the representation

$M_{t}(q)=\mathbb{E}[(\xi^{\scalebox{0.5}{$\mbox{\bf
L}$}}_{t})^{q}]=t\,(t+q)^{q-1}\;\,\mbox{remains valid if $q>-t$}.$

For more arguments justifying this extension on the Mellin transform, we refer
to Jedidi et al. JBH  for instance. Note that Pakes (pakes, , Theorem 3.4),
calculated only the moments of natural numbers order for
$\xi^{\scalebox{0.5}{$\mbox{\bf L}$}}_{1}$:

$\mathbb{E}[(\xi^{\scalebox{0.5}{$\mbox{\bf
L}$}}_{1})^{q}]=(1+q)^{q-1},\qquad\mbox{for}\;\,q=0,1,2,\cdots$

We can go beyond by observing that condition 3)(i) in Theorem 3.1 fails for
$\Psi_{\scalebox{0.5}{$\mbox{\bf L}$}}$. Nevertheless, since
$\mathfrak{L}[(\Psi_{\scalebox{0.5}{$\mbox{\bf L}$}})^{p},p+t]$ is finite for
every $p,\,t>0$, then (54) can be restated

$t\mapsto(t+p)\,\mathbb{E}[\xi_{t+p}^{-p}]=\frac{t+p}{t^{p+1}}=\frac{1}{t^{p}}+\frac{p}{t^{p+1}}\;\;\mbox{is
a completely monotone function.}$

Let $\phi$ be a Bernstein function, with inverse $\Psi$ and associated
subordinator $\xi$. Observe that it is not always possible to have $\Psi$
explicitly, for instance, take the Bernstein function
$\phi(\lambda)=\lambda/\log(1+\lambda),\;\lambda\geq 0$. Even one is lucky
enough to have $\Psi$ explicitly, it might be impossible to easily check
whether it is a Lévy-Laplace exponent! For instance, no standard calculation
would exhibit that
$\Psi(\lambda)=\lambda/[(1+\lambda)\log(1+1/\lambda)],\;\lambda\geq 0$, is
genuinely a Lévy-Laplace exponent, cf. (SSV, , Example 40 p. 320).

An answer to question (15) might be a consequence of Theorem 3.1: we show in
next result that we only need to strengthen the equivalence in Remark 4, $\xi$
needs to have a temporal Stieltjes property! Representation (56) below is
actually connected to the Laplace and Nevanlinna representations in (44). For

######  Corollary 1 (On temporally Stieltjes property for spectrally negative
Lévy processes)

  1. 1)

Let $\Psi$ be a Lévy-Laplace exponent represented by (38) and satisfying (52)
wiz. $\psi^{\prime}(0+)\in[0,\infty)$. Let the inverse function of (the non-
negative) function $\Psi$ be the Bernstein function $\phi$ associated to the
subordinator $\xi={(\xi_{t})}_{t\geq 0}$ given by (42) and (43) respectively.
Then, the following holds:

    1. (i)

For all $t>0$, $\xi_{t},$ satisfies the identity in law (45).

    2. (ii)

If $\,p\geq 0,\,$ or if either $\,0>p>-1\,$ and $\,\Psi^{p}\,$ is integrable
at 0, then the function $t\,\,\mathbb{E}[\xi_{t}^{-p}],\;t>0$, is well
defined, is completely monotone, and is represented by (46).

    3. (iii)

If $\,p\geq 1$ and $\Psi^{p}\in\mathcal{LE}(0,\infty)$, with quadruple of
characteristics $(0,\,\beta_{p},\,\gamma_{p},\,\nu_{p})$ in its representation
(38) and satisfying (52), then

$t\mapsto\frac{t^{2}}{\Gamma(p+1)}\,\mathfrak{L}[\Psi,t]=t\;\mathbb{E}[\xi_{t}^{-p}]\;\;\mbox{is
a Stieltjes transform,}$ (55)

and we have the representation

$\Gamma(p+1)\;t\;\mathbb{E}[\xi_{t}^{-p}]=\Psi^{\prime}(0+)+\frac{\gamma^{2}}{t}+\int_{(0,\infty)}\frac{x^{2}}{t+x}\,\nu_{p}(dx).$
(56)

The latter is equivalent to

$t\mapsto\frac{t^{3}}{\Gamma(p+1)}\,\mathfrak{L}[\Psi,t]=t^{2}\;\mathbb{E}[\xi_{t}^{-p}]\quad\mbox{is
a complete Bernstein function}.$ (57)

  2. 2)

The converse is stated as follows. Let $p\geq 1$ and ${(\xi^{(\scriptstyle
1/p)}_{t})}_{t\geq 0}$ be a $1/p$-standard stable subordinator, (see Example
2, with the convention $\xi^{(\scriptstyle 1)}_{t}=t$). If a subordinator
$\xi$ is such that the function $t\mapsto t\;\mathbb{E}[\xi_{t}^{-p}]$ is a
Stieltjes transform, then the subordinated process ${\big{(}\xi^{(\scriptstyle
1/p)}\circ\xi)_{t}\big{)}}_{t\geq 0}$ is the first passage time process of a
spectrally negative Lévy processes with Lévy-Laplace exponent
$\Psi^{p}\in\mathcal{LE}(0,\infty)$.

###### Remark 6

Last results merit the following comments:

  1. (i)

The equivalence between (56) and (57) is immediate by property (P6) above.

  2. (ii)

Recall the Lambert Bernstein function $\phi_{\scalebox{0.5}{$\mbox{\bf L}$}}$
given in Example 3 and its associated subordinator
$\xi^{\scalebox{0.5}{$\mbox{\bf L}$}}$. Since the inverse function
$\Psi_{\scalebox{0.5}{$\mbox{\bf L}$}}(x)=x\,e^{x}$ is not a Lévy-Laplace
exponent, then $\xi^{\scalebox{0.5}{$\mbox{\bf L}$}}$ is certainly not the
inverse time process of a Lévy process. Thus, Corollary 1 does not apply.

  3. (iii)

Assume we know that a subordinator $\xi={(\xi_{t})}_{t\geq 0}$ is such that
$\mathbb{E}[\xi_{t}^{-p}]$ is finite and is explicit for all $t>0$ and some
$p>0$, for example $\xi=\xi^{(\scriptstyle\alpha)}$ where is an
$\alpha$-stable subordinator. A “good test” to check whether $\xi$ is the
inverse time of some spectrally negative Lévy process with Lévy exponent
$\Psi$, is “just check if the temporal property (55) or (57) of $\xi$ holds
true. If it fails, then $\xi$ is certainly not an inverse time”.

  4. (iv)

Additionally, the operator applied on $\Psi$ in (57) has to be compared with
the one obtained by Schilling, Song and Vondraček (SSV, , Theorem 6.2.) and
illustrated by (33). Recall for instance the function
$\Psi_{\vartheta}(\lambda):=\lambda^{\vartheta},\;\lambda\geq 0$ used in
Example (1). $\Psi_{\vartheta}$ is a Lévy-Laplace exponent if, and only if,
$\vartheta\in[1,2]$ and then, $\Psi_{\vartheta}$ corresponds to a (non-
trivial) spectrally negative stable processes
$Z^{\vartheta}={(Z^{\vartheta}_{t})}_{t\geq 0}$ if, and only if
$\vartheta\in(1,2]$. Denoting $\alpha=1/\vartheta$, the subordinator
$\xi^{({\scriptscriptstyle\alpha})}={(\xi^{({\scriptscriptstyle\alpha})}_{t})}_{t\geq
0}$ is the inverse in time of $Z^{\vartheta}$ and is associated to the
Bernstein function $\lambda\mapsto\lambda^{\alpha},\,\alpha\in[1/2,1)$.
Observe that $\xi^{({\scriptscriptstyle\alpha})}$ has the so-called temporal
scaling property:
$\xi^{({\scriptscriptstyle\alpha})}_{t}\stackrel{{\scriptstyle
d}}{{=}}t^{1/\alpha}\,\xi^{({\scriptscriptstyle\alpha})}_{1}$. Example 2 gave,
that for $p>-\alpha$,

$\mathbb{E}\left[(\xi^{({\scriptscriptstyle\alpha})}_{t})^{-p}\right]=c\,t^{-\frac{p}{\alpha}},\quad
c:=\frac{\Gamma(\frac{p}{\alpha}+1)}{\Gamma(p+1)}$

and then

$t\mapsto
t^{2}\,\mathbb{E}\left[(\xi^{({\scriptscriptstyle\alpha})}_{t})^{-p}\right]=c\,t^{2-\frac{p}{\alpha}}\in\mathcal{CBF}\quad\Longleftrightarrow\lambda\mapsto(\Psi_{\vartheta})^{p}(\lambda)=\lambda^{p\vartheta}\in\mathcal{LE}(\mathds{R}+)$

and the latter is equivalent to $1\leq p\vartheta=p/\alpha\leq 2.$

##  4 Some results on Voiculescu transforms related to Hyperbolic functions

In the following, for Voiculescu (Nevanlinna) transforms found in Jurek j2 ,
we compute the triplet of characteristics $(\mathrm{a}_{{\scriptscriptstyle
X}},\,\mathrm{b}_{{\scriptscriptstyle
X}},\,\mathfrak{F}[\rho_{{\scriptscriptstyle X}},.])$ in their corresponding
representations (10) via the inversion formula (35). For more facts and
formulas, we refer to the Appendix at the end of this article. We recall that
$\tilde{X}$ indicates the free-probability analog, in the sense of the
procedure (P4) in Section 2, of the classical hyperbolic characteristic
function $X=C,\,S,T$ which are self-decomposable, that

$\mathfrak{F}_{{\scriptscriptstyle
C}}(s)=\frac{1}{\cosh(s)},\quad\mathfrak{F}_{{\scriptscriptstyle
S}}(s)=\frac{s}{\sinh(s)},\quad\mathfrak{F}_{{\scriptscriptstyle
T}}(s)=\frac{\tanh(s)}{s},\quad s\in\mathds{R}.$ (58)

and that the Voiculescu transforms of propositions 3, 4 and Corollary 2
correspond to the free-probability analogs $\tilde{X}$.

###### Proposition 3

Recall the $\beta$ function is defined in (grad, , 8.371(2)) by

$\beta(z)=\int_{0}^{\infty}e^{-zx}\,\frac{dx}{1+e^{-x}},\quad
z\in\mathds{C}^{+}.$ (59)

For the free-infinitely divisible Voiculescu transform

$V_{{\scriptscriptstyle\tilde{C}}}(iw)=-F_{{\scriptscriptstyle\tilde{C}}}(iw)=i\left[1-w\beta\left(\frac{w}{2}\right)\right],\quad
w>0,$

the characteristics are given by
$\mathrm{a}_{{\scriptscriptstyle\tilde{C}}}=\mathrm{b}_{{\scriptscriptstyle\tilde{C}}}=0$,
and the measure $\rho_{{\scriptscriptstyle\tilde{C}}}$ is a such that
$\rho_{{\scriptscriptstyle\tilde{C}}}(\mathds{R})=\frac{\pi}{2}-1$ and

$\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{C}}}\,;\,s]=2\,\sinh(s)\,\arctan\,(e^{-s})+\frac{\pi}{2}e^{-s}-1=\int_{0}^{\infty}\cos(sx)\,\frac{x}{(1+x^{2})\sinh\left(\frac{\pi}{2}x\right)}\,dx,\quad
s\in\mathds{R},$ (60)

where $\beta$ is given by (59) in the Appendix.

###### Proposition 4

Recall the digamma function is given by

$\psi(z)=\log
z+\int_{0}^{\infty}\big{(}\frac{1}{s}-\frac{1}{1-e^{-s}}\big{)}e^{-zs}ds,\quad
z\in\mathds{C}^{+},\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.361(8)})}$ (61)

and the Euler-Mascheroni constant corresponds to $\gamma:=-\psi(1)$. For the
free-infinitely divisible Voiculescu transform

$V_{{\scriptscriptstyle\tilde{S}}}(iw)=-F_{{\scriptscriptstyle\tilde{S}}}(iw)=i\left[w\,\psi\left(\frac{w}{2}\right)-w\,\log\left(\frac{w}{2}\right)+1\right],\quad
w>0,$

the characteristics are given by
$\mathrm{a}_{{\scriptscriptstyle\tilde{S}}}=\mathrm{b}_{{\scriptscriptstyle\tilde{S}}}=0$
and the measure $\rho_{{\scriptscriptstyle\tilde{S}}}$ is a such that
$\rho_{{\scriptscriptstyle\tilde{S}}}(\mathds{R})=\gamma+\log 2-1$ and

$\displaystyle\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{S}}}\,;\,s]$
$\displaystyle=$ $\displaystyle\frac{e^{-s}{\rm Ei}(s)+e^{s}{\rm
Ei}(-s)}{2}+\cosh(s)\,\log\left(\frac{1+e^{-|s|}}{1-e^{-|s|}}\right)-1$ (62)
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\cos(sx)\frac{x}{(1+x^{2})\left(e^{\pi
x}-1\right)}\,dx,\quad s\in\mathds{R}.$

where ${\rm Ei}$ is the exponential integral function given by (85) in the
Appendix.

By the elementary relation $\mathfrak{F}_{{\scriptscriptstyle
C}}=\mathfrak{F}_{{\scriptscriptstyle
S}}\cdot\mathfrak{F}_{{\scriptscriptstyle T}}$, we have
$C\stackrel{{\scriptstyle d}}{{=}}S+T$ , where $S$ and $T$ independent
versions. Using Remark 1, we can state:

###### Corollary 2

For the free-infinitely divisible Voiculescu transform,

$V_{{\scriptscriptstyle\tilde{T}}}(iw)=-F_{{\scriptscriptstyle\tilde{T}}}(iw)=F_{{\scriptscriptstyle\tilde{S}}}(iw)-F_{{\scriptscriptstyle\tilde{C}}}(iw)=iw\,\left[\,\log\left(\frac{w}{2}\right)-\beta\left(\frac{w}{2}\right)-\psi\left(\frac{w}{2}\right)\,\right],\quad
w>0,$

the characteristics are given by
$\mathrm{a}_{{\scriptscriptstyle\tilde{T}}}=\mathrm{b}_{{\scriptscriptstyle\tilde{T}}}=0$,
and the measure $\rho_{{\scriptscriptstyle\tilde{T}}}$ is a such that
$\rho_{{\scriptscriptstyle\tilde{T}}}(\mathds{R})=\frac{\pi}{2}-\gamma-\log 2$
and

$\displaystyle\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{T}}}\,;\,s]$
$\displaystyle=$
$\displaystyle\frac{\pi}{2}e^{-|s|}+2\sinh(s)\arctan(e^{-|s|})+\cosh(s)\,\log\frac{1-e^{-|s|}}{1+e^{-|s|}}-\frac{[e^{-s}{\rm
Ei}(s)+e^{s}{\rm Ei}(-s)]}{2}$ (63) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\cos(sx)\frac{2x}{(1+x^{2})\left(e^{\frac{\pi}{2}x}+1\right)}\,dx,\quad
s\in\mathds{R}.$

In general, if $X$ is a self-decomposable random variable, then the
corresponding Lévy measure $\nu_{{\scriptscriptstyle X}}$ has the form

$\nu_{{\scriptscriptstyle X}}(dx)=h_{{\scriptscriptstyle X}}(x)\,dx,\quad
x\neq 0$

$h$ is a measurable function such that $x\mapsto xh_{{\scriptscriptstyle
X}}(x)$ is increasing on $(-\infty,0)$ and decreasing on $(0,\infty)$.
Therefore, there exists a BDLP $Y=Y_{{\scriptscriptstyle X}}$, from the
corresponding random integral representations (16), such that the Lévy
measures is $\nu_{Y}(dx)=-(x\,h_{{\scriptscriptstyle X}})^{\prime}(x)\,dx$
whenever $h_{{\scriptscriptstyle X}}$ is differentiable, cf. Steutel and van
Harn (steutel, , Proposition 6.11 and Theorem 6.12, Chapter V) and also Jurek
(j1, , Corollary 1.1, p.97), (j2, , Section 2.1) or Jurek and Yor (jy, , p.
183, formulae (d) and (e)). Consequently, on the level of Nevanlinna measures,
we have, by (26)

$\rho_{\tilde{Y}_{{\scriptscriptstyle
X}}}(dx):=\frac{x^{2}}{1+x^{2}}\nu_{Y}(dx)=-\frac{x^{2}}{1+x^{2}}\,\big{(}h_{{\scriptscriptstyle
X}}(x)+xh_{{\scriptscriptstyle
X}}^{\prime}(x)\big{)}dx=-\rho_{{\scriptscriptstyle\tilde{X}}}(dx)-\frac{x^{3}\,h_{{\scriptscriptstyle
X}}^{\prime}(x)}{1+x^{2}}dx.$ (64)

where $\tilde{X}$ and $\tilde{Y}_{{\scriptscriptstyle X}}$ are the free
analogues of $X$ and $Y_{{\scriptscriptstyle X}}$, respectively. As in the
previous propositions we have a similar results for the BDLP’s as well,
although we computed it for $\tilde{Y}_{{\scriptscriptstyle C}}$, only:

###### Proposition 5

Let $K$ stands for the Catalan constant, and $\zeta$, $Li_{n}$ be the
Riemann’s zeta function and the polylogarithm functions given in (89) and (90)
in the Appendix. For the free-infinitely divisible Voiculescu transform

$V_{\tilde{Y}_{{\scriptscriptstyle C}}}(iw)=-F_{\tilde{Y}_{{\scriptscriptstyle
C}}}(iw)=i\,\left[1+\frac{w^{2}}{2}\,\zeta\left(2,\frac{w}{2}\right)-\frac{w^{2}}{4}\,\zeta\left(2,\frac{w}{4}\right)\right],\quad
w>0,$

the characteristics are given by $\mathrm{a}_{\tilde{Y}_{{\scriptscriptstyle
C}}}=\mathrm{b}_{\tilde{Y}_{{\scriptscriptstyle C}}}=0$ and the measure
$\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}$ is such that
$\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}(\mathds{R})=2\,K-1$ and

$\displaystyle\mathfrak{F}[\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}\,;\,s]$
$\displaystyle=$ $\displaystyle
2\,\cosh(s)\left(K-\int_{0}^{s}\frac{x}{\cosh(x)}dx\right)-s\tanh(s)-1$ (65)
$\displaystyle=$
$\displaystyle\frac{\pi}{2}\int_{0}^{\infty}\cos(sx)\,\frac{x^{2}}{1+x^{2}}\,\frac{\cosh(\frac{\pi
x}{2})}{\sinh^{2}(\frac{\pi x}{2})}dx,\quad s\in\mathds{R}.$

As a by-product of our Propositions 3 and 5, we have

###### Corollary 3

With the notations of Proposition 5, we have

$\mathfrak{F}[\rho_{\tilde{Y}_{{\scriptscriptstyle
C}}}\,;\,s]+\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{C}}}\,;\,s]=2\int_{0}^{\infty}\cos(sx)\frac{x^{3}}{1+x^{2}}\big{(}-h_{{\scriptscriptstyle
C}}(x)\big{)}^{\prime}\,dx,\quad s\in\mathds{R},$

where the function $h_{{\scriptscriptstyle
C}}(x):=1/\left(2x\sinh(\frac{\pi}{2}x)\right)$ is the density of the Lévy
measure of the hyperbolic cosine function $\mathfrak{F}_{{\scriptscriptstyle
C}}$.

###### Remark 7

Formula (60) is confirmed by (grad, , 4.113(8)) and (62) was confirmed
numerically for $s=0.5,\,1,\,2$. Formulae (63) and (65) seem to be new and
might be of some interest.

##  5 Proofs

###  5.1 Useful results on cumulant functions

Consider the class of cumulant functions denoted by

$\mathcal{CF}=\\{\lambda\mapsto\phi_{{\scriptscriptstyle
X}}(\lambda)=-\log\mathbb{E}[e^{-\lambda X}],\;\,\lambda\geq
0,\quad\mbox{where $X\,$ is a non-negative r.v.}\\}.$

###### Remark 8

The class $\mathcal{CF}$ contains the class of Bernstein functions
$\mathcal{BF}$. By injectivity of the Laplace transform, it is seen that to
every $\phi\in\mathcal{CF}$ corresponds a unique non-negative r.v. $X$ such
that $\phi=\phi_{{\scriptscriptstyle X}}$, i.e.

$\phi(\lambda)=-\log\mathbb{E}[e^{-\lambda
X}]=-\log\big{(}\mathbb{P}(X=0)+\mathbb{P}(X>0)\;\mathbb{E}[e^{-\lambda X}\mid
X>0]\big{)},\quad\lambda\geq 0.$ (66)

Observe that $\phi$ is linear if, and only if, $X$ is deterministic. Also
observe that

  * •

$\phi$ is infinitely differentiable $(0,\infty)$ and is a strictly increasing
bijection $[0,\infty)\to[0,l_{\phi})$. We denote from now on, by

$\Psi:[0,l_{\phi})\to\mathds{R}_{+},\;\,\mbox{the inverse function
of}\;\,\phi;$

  * •

we necessarily have

$l_{\phi}:=\lim_{\lambda\to\infty}{\phi(\lambda)}=-\log\mathbb{P}(X=0);$ (67)

  * •

if e denotes a r.v. with standard exponential distribution, then

$\mathbb{P}\left(\frac{\mbox{\bf
e}}{X}>x\right)=\mathbb{P}(X=0)+\mathbb{P}(X>0)\;\mathbb{E}[e^{-xX}\mid
X>0],\quad\mbox{for all $x\geq 0$}$

and

$\mathbb{P}\left(\Psi\big{(}\min(\mbox{\bf
e},l_{\phi})\big{)}>x\right)=\mathbb{P}\left(\min(\mbox{\bf
e},l_{\phi})>\phi(x)\right)=\mathbb{P}(\mbox{\bf
e}>l_{\phi})+\mathbb{P}\big{(}\mbox{\bf e}<l_{\phi},\mbox{\bf
e}>\phi(x)\big{)}=e^{-\phi(x)}.$

Exploiting last remarks, we propose the following lemma that will be useful in
the sequel:

######  Lemma 1 (Some results on cumulant functions)

Let $\phi:\mathds{R}_{+}\to\mathds{R}_{+}$ a differentiable function on
$(0,\infty)$, $\phi(0)=0$. Denote
$l_{\phi}:=\lim_{\lambda\to\infty}{\phi(\lambda)}$ and by e a r.v. with
standard exponential distribution.

  1. 1)

Assume $\phi$ is a non-linear cumulant function, hence is associated to a non-
negative and non-deterministic r.v. $X$. Then,

    1. (i)

$\phi^{\prime}$ is a (strictly) decreasing bijection on $(0,\infty)$, hence
$\phi$ is strictly concave and increases to $l_{\phi}$;

    2. (ii)

The inverse function $\Psi$ of $\phi$, is such that
$\Psi^{\prime}:(0,l_{\phi})\to(0,\infty)$ is a (strictly) increasing
bijection, hence $\Psi$ is strictly convex;

    3. (iii)

As $\lambda\uparrow\infty$, the function $\phi(\lambda)/\lambda$ decreases to

$L_{X}:=\inf\\{x\geq 0,\;\mbox{s.t.}\;\mathbb{P}(X\leq x)>0\\}.$

Hence, on $(0,\infty)$, the function $\Psi(x)/x$ increases to $1/L_{X}$ as
$x\uparrow l_{\phi}$. Further,

$L_{X}=\lim_{x\to\infty}\phi^{\prime}(x)\in[0,\infty);$ (68)

    4. (iv)

If $p>0$, then $(\Psi^{p})^{\prime}$ is positive on $(0,\infty)$ and we have
the equivalence

$(\Psi^{p})^{\prime}(0+):=\lim_{\lambda\to
0+}(\Psi^{p})^{\prime}(\lambda)<\infty\Longleftrightarrow\lim_{\lambda\to
0+}\frac{\lambda^{p-1}}{\phi^{\prime}(\lambda)};$ (69)

    5. (v)

The function $\Psi^{p}$, is convex (respectively concave) if, and only if,

$\lambda\mapsto\lambda^{1-p}\,\phi^{\prime}\;\mbox{is non-increasing
(respectively non-decreasing )}.$

In particular, if $p\geq 1$, then $\Psi^{p}$ is strictly convex and
$(\Psi^{p})^{\prime}(0+)\in[0,\infty)$.

  2. 2)

Assume $\phi$ is a strictly increasing bijection with inverse $\Psi$. Then,
the following statements are equivalent.

    1. i)

$\phi$ is a cumulant function;

    2. ii)

For some non-negative r.v. $X$, we have the identity in law

$\frac{\mbox{\bf e}}{X}\stackrel{{\scriptstyle
d}}{{=}}\Psi\left(\min\left(\mbox{\bf
e},l_{\phi}\right)\right),\qquad\mbox{(where $\mbox{\bf e}$ in the l.h.s. is
assumed to be independent of $X$};$ (70)

    3. iii)

For some non-negative r.v. $X$, we have the representations

$\mathbb{E}[X^{-p}]=\frac{M(p)}{\Gamma(p+1)},\qquad
M(p):=\int_{0}^{l_{\phi}}e^{-x}\,\Psi^{p}(x)\,dx+e^{-l_{\phi}}\,\Psi^{p}(l_{\phi})\,{\rm
1\leavevmode\nobreak\ l}_{(l_{\phi}<\infty)},$ (71)

for every $p>-1,\,$ such that $\,e^{-x}\,\Psi^{p}(x)\,$ is integrable on
$(0,l_{\phi})$.

###### Remark 9

Observe that formula (71) reminds the one involving the Gamma function in
(84). Actually, under the condition of integrability of $\Psi^{p}$, the
injectivity of the Mellin transform insures that the function $M(p),\;p>-1$ is
the Mellin transform of a (unique) positive random variable, namely

$M(p)=\mathbb{E}\left[\Psi^{p}\big{(}\min(\mbox{\bf
e},l_{\phi})\big{)}\right],$ (72)

thus, (71) reads

$\mathbb{E}\left[\,\left(\frac{\mbox{\bf
e}}{X}\right)^{p}\,\right]=\mathbb{E}[\mbox{\bf
e}^{p}]\;\mathbb{E}[X^{-p}]=\Gamma(p+1)\;\mathbb{E}[X^{-p}]=\mathbb{E}\left[\Psi^{p}\big{(}\min(\mbox{\bf
e},l_{\phi})\big{)}\right].$

Proof of Lemma 1. 1) $(i)$: The first and the second derivative of $\phi$ are
given by

$\phi^{\prime}(\lambda)=\frac{\mathbb{E}[Xe^{-\lambda
X}]}{\mathbb{E}[e^{-\lambda X}]}\quad\mbox{and
}\quad\phi^{\prime\prime}(\lambda)=\frac{\mathbb{E}[Xe^{-\lambda
X}]^{2}-\mathbb{E}[X^{2}e^{-\lambda X}]\mathbb{E}[e^{-\lambda
X}]}{\mathbb{E}[e^{-\lambda X}]^{2}},\qquad\lambda>0.$ (73)

By Cauchy-Schwarz inequality, every pair of nonnegative random variable $Y,Z$
satisfy the inequality
$\mathbb{E}[yZ]^{2}\leq\mathbb{E}[Y^{2}]\mathbb{E}[Z^{2}]$ and the equality
holds if and only if there exists $c\geq 0$ such that $Y=cZ$ a.s. Choosing
$Y=Xe^{-\frac{\lambda}{2}X}$ and $Z=e^{-\frac{\lambda}{2}X}$ in (73), we
recover the negativity of $\phi^{\prime\prime}$ and the claim on
$\phi^{\prime}$ follows.
1) $(ii)$: The assertion is due to the observation
$\Psi^{\prime}=1/\phi^{\prime}(\Psi)$ and
$\Psi^{\prime\prime}=-\phi^{\prime\prime}(\Psi)/(\phi^{\prime}(\Psi))^{3}$.
1) $(iii)$ The assertion is an adaptation of the result of Pakes (pakesc, ,
Theorem 2.1) by taking the Mellin transform
$M(\lambda)^{1/\lambda}=\mathbb{E}[(e^{-X})^{\lambda}]^{1/\lambda}=e^{-\phi(\lambda)/\lambda}$
there. The limit in (68) is obtained by Karamata’s Theorem (BIN, , Theorem
1.5.11), the monotone density theorem (BIN, , Theorem 1.7.2) and the concavity
of a cumulant function.
1) $(iv)$: We have

$(\Psi^{p})^{\prime}=p\;\frac{\Psi^{p-1}}{\phi^{\prime}\left(\Psi\right)},\qquad(\Psi^{p})^{\prime\prime}=\frac{p\;\Psi^{p-2}}{\phi^{\prime}(\Psi)^{3}}\left[(p-1)\,\phi^{\prime}(\Psi)-\Psi\phi^{\prime\prime}(\Psi)\right]$
(74)

Positivity of $(\Psi^{p})^{\prime}$ is immediate and since $\Psi(0)=0$, then
by the change of variable $x=\phi(\lambda)$, we have

$\lim_{x\to 0+}(\Psi^{p})^{\prime}(x)=p\;\lim_{x\to
0+}\frac{\Psi^{p-1}(x)}{\phi^{\prime}(\Psi(x))}=p\;\lim_{\lambda\to
0+}\frac{\lambda^{p-1}}{\phi^{\prime}(\lambda)}.$ (75)

For a later use, one can observe that

$p\geq 1\Longrightarrow\lim_{x\to
0+}x\;(\Psi^{p})^{\prime}(x)=p\;\lim_{\lambda\to
0+}\phi(\lambda)\frac{\lambda^{p-1}}{\phi^{\prime}(\lambda)}\in[0,\infty).$
(76)

1) $(v)$: By (74), we have

$Sign\left((\Psi^{p})^{\prime\prime}\right)=Sign\left[(p-1)\,\phi^{\prime}-\lambda\phi^{\prime\prime}\right]=Sign\left[\left(\frac{\lambda^{p-1}}{\phi^{\prime}}\right)^{\prime}\right]$

and if $p\geq 1$, then $\lambda^{p-1}/\phi^{\prime}$ is increasing, viz.
$\Psi^{p}$ is convex and last assertion is then obtained with the help of
(75).
2) $(i)\Longleftrightarrow(ii)$: Is obtained from the discussion in Remark 8.
2) $(ii)\Longrightarrow(iii)$: The case $l_{\phi}=\infty$ is easy to prove due
to the facts, that $\Psi(0)=\phi(0)=0$, $\Psi$ is an increasing bijection from
$\mathds{R}_{+}\to\mathds{R}_{+}$ (with inverse $\phi$), and that

$-\frac{d}{dy}\mathbb{E}[e^{-yX}]=-\frac{d}{dy}e^{-\phi(y)}=\phi^{\prime}(y)\,e^{-\phi(y)}=\mathbb{E}[X\,e^{-yX}].$

Indeed, making the change of variable $x=\phi(y)$, using Tonelli-Fubini’s
theorem and the representation of the gamma function (84), these facts entail

$\displaystyle\int_{0}^{\infty}e^{-x}\,\frac{\Psi^{p}(x)}{\Gamma(p+1)}\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\phi^{\prime}(y)\,e^{-\phi(y)}\,\frac{y^{p}}{\Gamma(p+1)}\,dy=\int_{0}^{\infty}\mathbb{E}[X\,e^{-yX}]\,\frac{y^{p}}{\Gamma(p+1)}\,dy$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{1}{X^{p}}\underbrace{\int_{0}^{\infty}\,e^{-yX}\,\frac{X^{p+1}\,y^{p}}{\Gamma(p+1)}\,dy}_{=1}\right]=\mathbb{E}\left[X^{-p}\right].$

The case $l_{\phi}<\infty$ is proved similarly.
2) $(iii)\Longrightarrow(ii)$: This has been noticed in Remark 9. $\square$

###  5.2 The proofs

Proof of Theorem 2.1. Observe that if $F$ is a Nevanlinna-Pick function
represented by (5), then $\mathrm{a},\,\mathrm{b}$ and $\rho(\mathds{R})$ are
given by (7).
1) $\Longrightarrow$ 2): It is enough to reproduce the steps of the proof
(SSV, , Corollary 6.13) taking into account that $\rho$ is supported by $A$.
2) $\Longrightarrow$ 3): Use Jankowski and Jurek inversion’s procedure given
(jan, , Theorem 1) and explained before (10) and observe that
$H(w)=G(iw),\;w>0$. Also observe that

$\mathfrak{L}\big{[}\sinh(x);w]=\frac{1}{w^{2}-1}\quad\mbox{and}\quad\mathfrak{L}[\cosh(x);w]=\frac{w}{w^{2}-1},\quad
w>1,$ (77)

which yields (35).
3) $\Longrightarrow$ 1): Observe that knowing the function $w\mapsto
F(iw)=i(1-w^{2})\,H(w)$ on the interval $(1,\infty)$ or even on any interval
$(a,b)\subset(0,\infty)$ is sufficient to fully recover $F$, cf. Jedidi et al.
(JBH, , Lemma 4.1).

The last claim is a straightforward adaptation of (SSV, , Corollary 6.13).
$\square$

Proof of Proposition 1. In order to get (44), just use the representation (38)
of $\Psi$, the properties (84) of the Gamma function and the identity

$-z^{2}\left(\frac{1}{z+x}-\frac{1}{z}+\frac{1}{z^{2}}\frac{x}{1+x^{2}}\right)=\frac{zx-1}{z+x}\,\frac{x^{2}}{1+x^{2}},$

in order to write: for $z\in\mathds{C}^{+}$

$\displaystyle F(z)$ $\displaystyle=$ $\displaystyle
z^{2}\int_{0}^{\infty}e^{-zu}\,\left(\mathrm{b}-\beta
u-\frac{\gamma^{2}}{2}u^{2}-\int_{(0,\infty)}\left(e^{-ux}-1+u\frac{x}{1+x^{2}}\right)\nu(dx)\right)\,du$
$\displaystyle=$
$\displaystyle-\beta+\mathrm{b}z-\frac{\gamma^{2}}{z}-z^{2}\int_{(0,\infty)}\left(\frac{1}{z+x}-\frac{1}{z}+\frac{1}{z^{2}}\frac{x}{1+x^{2}}\right)\nu(dx)$
$\displaystyle=$
$\displaystyle-\beta+\mathrm{b}z+\int_{[0,\infty)}\frac{zx-1}{z+x}\,\left(\gamma^{2}\delta_{0}(dx)+\frac{x}{1+x^{2}}\nu(dx)\right)$
$\displaystyle=$
$\displaystyle\mathrm{a}+\mathrm{b}z+\int_{[0,\infty)}\frac{zx-1}{z+x}\rho(dx).$

The controls (25) and Tonelli-Fubini’s theorem allow the reversal of order in
the integrals. $\square$

Proof of Proposition 2. 1) $\Longrightarrow$ 2): If $\phi\in\mathcal{BF}$,
then it is associated to a subordinator $\xi={(\xi_{t})}_{t\geq 0}$, such that
each $\xi_{t},\;t>0$, has the cumulant (Bernstein) function $\phi_{t}=t\,\phi$
whose inverse is $\Psi_{t}(x)=\Psi(x/t)$. Since $\phi(\infty)=\infty$, then
assertion (45) immediately follows from (70).
2) $\Longrightarrow$ 3): is trivially satisfied with $\eta=\xi$.
3) $\Longrightarrow$ 1): By 2) in Lemma 1, necessarily the inverse $t\phi$ of
$\Psi_{t}(x)=\Psi(x/t)$ is the cumulant of $\eta_{t}$ for all $t>0$, i.e.,
$e^{-t\phi}$ is a completely monotone, or equivalently $(1-e^{-t\phi})/t$ is a
Bernstein function for all $t>0$. Passing to the limit as $t\to 0$ and using
the fact that the class $\mathcal{BF}$ is closed under pointwise limits, cf.
(SSV, , Corollary 3.8), we deduce that $\phi\in\mathcal{BF}$.
3) $\Longrightarrow$ 4): By the implication 2) $\Longrightarrow$ 3), point 3)
is equivalent to the existence of the subordinator $\xi$ satisfying (45), then
(46) follows from of (71) and from the change of variable $x\rightarrow tx$
there.
4) $\Longrightarrow$ 1): By Lemma 1, necessarily the inverse $t\phi$ of
$\Psi_{t}(x)=\Psi(x/t)$ is the cumulant of $\xi_{t}$ and $\xi$ is a
subordinator. Then, $\phi$ is a Bernstein function. $\square$

Proof of Theorem 3.1. 1) Property 1) $(iii)$ in Lemma 1 gives the implication

$x\mapsto\frac{\Psi(x)}{x}\;\mbox{is non-decreasing},\quad
p\in(-1,0)\Longrightarrow e^{-tx}\Psi^{p}(x)\;\mbox{is integrable at
$\infty$,}\quad\forall t>0.$

Then, the definiteness of $\mathfrak{L}[\Psi^{p}\,;\,.]$ on $(0,\infty)$ is
equivalent to 1)$(i)$. Since $\Psi^{p}$ is a non-increasing function, the
latter is equivalent to assertion 1)$(ii)$, because representation (46) of
$\mathbb{E}[\xi_{t}^{-p}]$ meets the form (SSV, , (3.3) p. 23) of a Bernstein
function. Representation (48) is obtained by integration by parts fom (46).
2) Similarly,

$x\mapsto\frac{\Psi(x)}{x}\;\mbox{is non-decreasing},\quad p\geq
0\Longrightarrow\Psi^{p}(x)\;\mbox{is integrable at}\;0.$

Then, the definiteness of $\mathfrak{L}[\Psi^{p}\,;\,.]$ $(0,\infty)$ is then
equivalent to 2)$(i)$. Since $\Psi^{p}$ is an non-decreasing function, null at
0, and by an integration by parts, the latter is equivalent to representation
(49), hence $\mathbb{E}[\xi_{t}^{-p}]$ meets the form of a completely monotone
function. The last claim is due to 1) $(v)$ in Lemma 1 that insures the
concavity of $\Psi^{p}$, and to the form (46) that gives

$t\;\mathbb{E}[\xi_{t}^{-p}]=\frac{t^{2}}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\;\Psi^{p}(x)\,dx,$
(78)

which meets the form (SSV, , (3.4) p. 23) of a Bernstein function.
Representation (50) is obtained by integration by parts in (49).
3) The convexity of $\Psi^{p}$ and $(\Psi^{p})^{\prime}(0+)\in[0,\infty)$ are
guaranteed by 1)$(v)$ in Lemma 1. The integrability of $\Psi^{p}$ near zero is
guaranteed by $p\geq 1$. If further $\mathfrak{L}[\Psi^{p}\,;\,t]<\infty$, for
some $t>0$, then two integration by parts in (78), give representation

$t\;\mathbb{E}[\xi_{t}^{-p}]\;=\frac{1}{\Gamma(p+1)}\left((\Psi^{p})^{\prime}(0)+\int_{0}^{\infty}e^{-tx}\,(\Psi^{p})^{\prime\prime}(x)\,dx\right)$
(79)

Differentiation

$-\frac{d}{dt}\left(t\;\mathbb{E}[\xi_{t}^{-p}]\right)=\frac{1}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\;x\;(\Psi^{p})^{\prime\prime}(x)\,dx.$

The function $t\;\mathbb{E}[\xi_{t}^{-p}]$ being positive, deduce the
equivalences $3)(i)\Longleftrightarrow 3)(ii)$ from $2)(i)\Longleftrightarrow
2)(ii)$.

The last claims in 3) are obtained as follows:
3) $(a)$ Representation (49), when multiplied by $t^{2}$ meets with the form
(SSV, , (3.4) p. 23) of a Bernstein function, if, and only if
$(\Psi^{p})^{\prime}$ is concave.
3) $(b)$ Representation (53) is a simple reformulation of (51), when
differentiating twice the expression of $\Psi^{p}$ in (38). $\square$

Proof of Corollary 1. 1)$(i)$ and 1)$(ii)$ are a direct consequence of
Proposition 2 because $\xi$ is a subordinator.
1)$(iii)$: If $\Psi^{p}\in\mathcal{LE}(0,\infty)$, then, by Proposition 2, the
function $f(t):=t\,\mathbb{E}[\xi_{t}^{-p}],\;t>0,$ has the representation

$f(t)=\frac{t^{2}}{\Gamma(p+1)}\int_{0}^{\infty}e^{-tx}\,\Psi^{p}(x)\,dx.$

By Proposition 1, the extension on $\mathds{H}^{+}$ of $f$ is such that $-f$
is a non-positive Nevanlinna $\mathcal{N}(\mathds{R}_{+})$-function, which is
continuous on $(0,\infty)$. By (SSV, , Corollary 6.14) this equivalent to say
that $f$ is a Stieltjes transform and by (SSV, , Theorem 6.2), this also
equivalent to $t\mapsto t\,f(t)\in\mathcal{CBF}$.
2) Since $\xi$ is a subordinator, then, by Proposition 2, the function
$t\mapsto f(t)=:t\;\mathbb{E}[\xi_{t}^{-p}]$ is represented by

$-f(t)=-t^{2}\int_{0}^{\infty}e^{-tx}\,\Psi^{p}(x)\,dx.$

Since $f$ is a Stieltjes function, then, using (SSV, , Corollary 6.14) again,
we deduce the extension on $\mathds{H}^{+}$ of $-f$ is a Nevanlinna
$\mathcal{N}^{\star}(\mathds{R}_{+})$-function because it satisfies
$\lim_{|z|\to\infty,z\in\mathds{C}^{+}}-f(z)/z=0$. Finally, by Proposition 1,
we necessarily have $\Psi^{p}\in\mathcal{LE}(0,\infty)$. To conclude, observe
that the inverse function of $\Psi^{p}$ is the Bernstein function
$(\Psi^{p})^{-1}(\lambda)=\phi(\lambda^{1/p})$ corresponding to the
subordinator $\xi^{(\scriptstyle 1/p)}\circ\xi$. $\square$

In all the following proofs, the symbol $\ast$ denotes the usual additive
convolution of functions on the positive half-line. The calculi where also
checked by WolframAlpha or Mathematica.

Proof of Proposition 3. Firstly, note that
$F_{{\scriptscriptstyle\tilde{C}}}(i)=-i\big{(}1-\beta(1/2)\big{)}=i(\pi/2-1)$
and therefore
$\rho_{{\scriptscriptstyle\tilde{C}}}(\mathds{R})=\frac{\pi}{2}-1$. Secondly,
by (59), we have
$\beta(z)=\mathfrak{L}[(1+e^{-x})^{-1};z],\;z\in\mathds{C}^{+}.$ Consequently,

$\displaystyle\frac{iF_{{\scriptscriptstyle\tilde{C}}}(iw)}{w^{2}-1}$
$\displaystyle=$
$\displaystyle\frac{1-w\beta(\frac{w}{2})}{w^{2}-1}=\mathfrak{L}[\sinh(s);w]-\mathfrak{L}[\cosh(s)\,;w\,]\;\mathfrak{L}\left[\frac{2}{1+e^{-2s}}\,;w\,\right]$
(80) $\displaystyle=$
$\displaystyle\mathfrak{L}[\sinh(s);w]-\mathfrak{L}\left[\,\left(\cosh(u)\ast\frac{2}{1+e^{-2u}}\right)(s);w\right],\quad\mbox{for}\;\;w>0,\,w\neq
1.$

Thirdly, by a differentiation, one checks that, for $s>0$,

$\left(\cosh(u)\ast\frac{2}{1+e^{-2u}}\right)(s):=\int_{0}^{s}\cosh(s-u)\,\frac{2}{1+e^{-2(s-u)}}du=2\sinh(s)\,\left[\frac{\pi}{4}-\arctan(e^{-s})\right]-e^{-s}+1,$

and inserting the latter into (80), we get

$\frac{iF_{{\scriptscriptstyle\tilde{C}}}(iw)}{w^{2}-1}=\mathfrak{L}\left[\sinh(s)-2\sinh(s)\left(\frac{\pi}{4}-\arctan(e^{-s})\right)+e^{-s}-1\,;w\,\right].$
(81)

Finally, since
$\rho_{{\scriptscriptstyle\tilde{C}}}(\mathds{R})=\frac{\pi}{2}-1$ and using
(77), we arrive to

$\displaystyle\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{C}}}\,;\,s]$
$\displaystyle=$
$\displaystyle\left(\frac{\pi}{2}-1\right)\cosh(s)+\sinh(s)-2\sinh(s)\,\left(\frac{\pi}{4}-\arctan(e^{-s})\right)+e^{-s}-1$
$\displaystyle=$ $\displaystyle
2\,\sinh(s)\,\,\arctan(e^{-s})+\frac{\pi}{2}e^{-s}-1,$

which gives the first equality in (60). On the other hand, from Jurek (j2, ,
Corollary 2), we know that
$V_{{\scriptscriptstyle\tilde{C}}}(iw)=-F_{{\scriptscriptstyle\tilde{C}}}(iw)$
is a free-probability analog of the classical hyperbolic characteristic
function $1/\cosh(t)$ whose Nevanlinna measure
$\rho_{{\scriptscriptstyle\tilde{C}}}$ has a density

$\frac{|x|}{2(1+x^{2})\sinh(\frac{\pi}{2}|x|)},\quad x\in\mathds{R}.$

Thus,

$\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{C}}}\,;\,s]=\int_{\mathds{R}}e^{isx}\frac{|x|}{2(1+x^{2})\sinh(\frac{\pi}{2}|x|)}\,dx=\int_{0}^{\infty}\cos(sx)\frac{x}{(1+x^{2})\sinh(\frac{\pi}{2}x)}\,dx,$

which shows the second equality in (60). $\square$

Proof of Proposition 4. By (grad, , 8.366(2)), we have
$\psi(1/2)=-\gamma-2\log 2$, then
$F_{{\scriptscriptstyle\tilde{S}}}(i)=i(\gamma+\log 2-1)$. Hence, in (10), we
have
$\mathrm{a}_{{\scriptscriptstyle\tilde{S}}}=\mathrm{b}_{{\scriptscriptstyle\tilde{S}}}=0$
and the measure $\rho_{{\scriptscriptstyle\tilde{S}}}$ has a finite mass
$\rho_{{\scriptscriptstyle\tilde{S}}}(\mathds{R})=\gamma+\log 2-1$. Using the
integral formula (61) for the $\psi$-function, we get

$\displaystyle\frac{iF_{{\scriptscriptstyle\tilde{S}}}(iw)}{w^{2}-1}$
$\displaystyle=$
$\displaystyle\frac{1+w\,(\psi(w/2)-\log(w/2))}{w^{2}-1}=\mathfrak{L}[\sinh(s);w]+\mathfrak{L}[\cosh(s);w]\;\mathfrak{L}\left[\frac{1}{s}-\frac{2}{1-e^{-2s}};w\right]$
(82) $\displaystyle=$
$\displaystyle\mathfrak{L}\left[\sinh(s)+g(s);w\right],\quad\mbox{where}\;g(s):=\left(\cosh(u)\ast\big{(}\frac{1}{u}-\frac{2}{1-e^{-2u}}\big{)}\right)(s).$

By a direct differentiation in identities (F3) in the Appendix of the ${\rm
Ei}$ function, one checks that, for $s>0$, we have

$\displaystyle g(s)$ $\displaystyle=$
$\displaystyle\int_{0}^{s}\cosh(s-u)\left[\frac{1}{u}-\frac{2}{1-e^{-2u}}\right]\,du=\left[\frac{e^{-s}}{2}{\rm
Ei}(u)+\frac{e^{s}}{2}{\rm
Ei}(-u)-e^{u-s}+\cosh(s)\,\log\frac{1+e^{-u}}{1-e^{-u}}\right]\Big{|}^{u=s}_{u=0^{+}}$
$\displaystyle=$ $\displaystyle\frac{e^{-s}}{2}{\rm Ei}(s)+\frac{e^{s}}{2}{\rm
Ei}(-s)-1+\cosh(s)\,\log\frac{1+e^{-s}}{1-e^{-s}}-\lim_{u\to
0^{+}}\Big{[}\frac{e^{-s}}{2}({\rm Ei}(u)-\log(1-e^{-u}))$
$\displaystyle\quad+\frac{e^{s}}{2}({\rm
Ei}(-u)-\log(1-e^{-u}))-e^{u-s}+\cosh(u)\log(1+e^{-u})\big{]}$
$\displaystyle=$ $\displaystyle\frac{e^{-s}}{2}{\rm Ei}(s)+\frac{e^{s}}{2}{\rm
Ei}(-s)-1+e^{-s}+\cosh(s)\left(\log\frac{1+e^{-s}}{1-e^{-s}}-\gamma-\log
2\right).$

Inserting the above equality into (82) and using (77) with
$\rho_{{\scriptscriptstyle\tilde{S}}}(\mathds{R})=\gamma+\log 2-1$, we get for
$s\in\mathds{R}$,

$\displaystyle\mathfrak{F}[\rho_{{\scriptscriptstyle\tilde{S}}}\,;\,s]$
$\displaystyle=$ $\displaystyle(\gamma+\log
2-1)\cosh(s)+\sinh(s)+\frac{e^{-s}}{2}{\rm Ei}(s)+\frac{e^{s}}{2}{\rm
Ei}(-s)-1+e^{-s}+\cosh(s)\left(\log\frac{1+e^{-|s|}}{1-e^{-|s|}}-\gamma-\log
2\right)$ $\displaystyle=$ $\displaystyle\frac{e^{-s}}{2}{\rm
Ei}(s)+\frac{e^{s}}{2}{\rm
Ei}(-s)-1+\cosh(s)\,\log\frac{1+e^{-|s|}}{1-e^{-|s|}},$

which proves the equality (62). From Jurek (j2, , Corollary 3), we know that
$V_{{\scriptscriptstyle\tilde{S}}}(iw)=-F_{{\scriptscriptstyle\tilde{S}}}(iw)$
is the free-analog of the classical hyperbolic sine characteristic function
$\mathfrak{F}_{{\scriptscriptstyle S}}(t)=t/\sinh(t)$ whose Nevanlinna measure
equals to

$\rho_{{\scriptscriptstyle\tilde{S}}}(dx)=\frac{1}{2}\frac{|x|}{1+x^{2}}\frac{e^{-\frac{\pi}{2}|x|}}{\sinh(\frac{\pi}{2}|x|)}dx=\frac{|x|}{1+x^{2}}\,\frac{1}{e^{\pi|x|}-1}dx,\quad
x\in\mathds{R},$

and we get the equality (62). $\square$

Proof of Proposition 5. Since $F_{\tilde{Y}_{{\scriptscriptstyle
C}}}(i)=i(2K-1)$, then in (10), $\mathrm{a}_{\tilde{Y}_{{\scriptscriptstyle
C}}}=\mathrm{b}_{\tilde{Y}_{{\scriptscriptstyle C}}}=0$, and for the measure
$\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}$, we have
$\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}(\mathds{R})=2K-1$. Using (77) and
the integral representation (91) for the $\zeta(2,s)$ function, we have

$\displaystyle\mathfrak{L}[\mathfrak{F}[\rho_{\tilde{Y}_{{\scriptscriptstyle
C}}}](s)-(2K-1)\cosh(s)\,;w\,]$ $\displaystyle=$
$\displaystyle\frac{1}{w^{2}-1}\,\left[\frac{w^{2}}{2}\,\left(\zeta(2,\frac{w}{2})-\frac{1}{2}\,\zeta(2,\frac{w}{4})\right)+1\right]$
$\displaystyle=$
$\displaystyle\frac{1}{w^{2}-1}+2\frac{w^{2}}{w^{2}-1}\left[\int_{0}^{\infty}e^{-wu}\,\frac{u}{1-e^{-2u}}\,du-2\int_{0}^{\infty}e^{-wu}\,\frac{u}{1-e^{-4u}}\,du\right]$
$\displaystyle=$
$\displaystyle\mathfrak{L}[\sinh(s);w]+2\left(1+\frac{1}{w^{2}-1}\right)\,\int_{0}^{\infty}e^{-wu}\,u\;\frac{e^{-2u}-1}{1-e^{-4u}}\,du$
$\displaystyle=$
$\displaystyle\mathfrak{L}[\sinh(s);w]+(1+\frac{1}{w^{2}-1})\int_{0}^{\infty}e^{-wu}\,u\;\frac{1-e^{2u}}{\sinh(2u)}\,du$
$\displaystyle=$
$\displaystyle\mathfrak{L}\left[\sinh(s)+s\frac{1-e^{2s}}{\sinh(2s)};w\right]+\mathfrak{L}[\sinh(s)\,;w\,]\;\mathfrak{L}\left[s\frac{1-e^{2s}}{\sinh(2s)}\,;w\,\right]$
$\displaystyle=$
$\displaystyle\mathfrak{L}\left[\sinh(s)+s\frac{1-e^{2s}}{\sinh(2s)}+h(s);w\right],$

where,

$h(s):=(\sinh u\ast
u\frac{1-e^{2u}}{\sinh(2u)})(s)=\int_{0}^{s}\sinh(s-u)\,u\;\frac{1-e^{2u}}{\sinh(2u)}\,du.$
(83)

Using elementary computations, together with (88), (93) in the Appendix, one
checks that

$\displaystyle h(s)$ $\displaystyle=$
$\displaystyle\left[e^{-s}(u-1)\,e^{u}-i\cosh(s)\,\left({\rm
Li}_{2}(ie^{u})-{\rm
Li}_{2}(-ie^{u})+u\,\log\frac{1-ie^{u}}{1+ie^{u}}\right)\right]_{u=0}^{u=s}$
$\displaystyle=$
$\displaystyle\left[e^{-s}(u-1)\,e^{u}-i\cosh(s)\,\left(-2i\int_{0}^{u}\frac{x}{\cosh(x)}dx\right)\right]_{u=0}^{u=s}$
$\displaystyle=$ $\displaystyle
e^{-s}-1+s-2\cosh(s)\,\int_{0}^{s}\frac{x}{\cosh(x)}dx.$

From the above calculation and from the equality
$(e^{2s}-1)/\sinh(2s)=2e^{2s}/(e^{2s}+1)$, we arrive at the representation of
$\mathfrak{F}[\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}],\;s\geq 0$:

$\displaystyle\mathfrak{F}[\rho_{\tilde{Y}_{{\scriptscriptstyle C}}}](s)$
$\displaystyle:=$
$\displaystyle(2K-1)\cosh(s)+\sinh(s)-2s\,\frac{e^{2s}}{e^{2s}+1}+h(s)$
$\displaystyle=$ $\displaystyle
2\,\cosh(s)\left(K-\int_{0}^{s}\frac{x}{\cosh(x)}dx\right)+s\left(1-2\frac{e^{2s}}{e^{2s}+1}\right)+\overbrace{\big{(}\sinh(s)-\cosh(s)+e^{-s}\big{)}}^{=0}-1$
$\displaystyle=$ $\displaystyle
2\,\cosh(s)\left(K-\int_{0}^{s}\frac{x}{\cosh(x)}dx\right)-s\tanh(s)-1$

which gives the equality (65). For the equality (65), we just need to recall
that from Jurek and Yor (jy, , Corollary 1 and a formula (7) p. 186), that the
Nevanlinna measure corresponding to the BDLP $Y_{{\scriptscriptstyle C}}$,
given by (16) is represented by

$\rho_{\tilde{Y}_{{\scriptscriptstyle
C}}}(dx)=\frac{\pi}{4}\,\frac{x^{2}}{1+x^{2}}\,\frac{\cosh\left(\frac{\pi
x}{2}\right)}{\sinh^{2}\left(\frac{\pi x}{2}\right)}dx,\quad x\in\mathds{R}.$

$\square$

Proof of Corollary 3. It is a simple application of Remark (64) applied to
$X=C$. $\square$

###### Remark 10

_Since the hyperbolic sine and the hyperbolic tangent are self-decomposable as
well, we may have a statement about $\tilde{S}$ and $\tilde{T}$, analogous to
the one in Corollary 3 for the hyperbolic cosine function._

##  6 Appendix

For a convenience of reading, we collect some facts on special functions.
Boldface numbers below refer to formulae from grad .

  1. (F1)

Many of those functions we use are derived from _Euler’s $\Gamma$ gamma
function_ and the _digamma function_ $\psi$ used in Proposition 4:

$\frac{\Gamma(p)}{z^{p}}:=\,\int_{0}^{\infty}e^{-zx}\,x^{p-1}\,dx,\quad\psi(p):=\frac{\Gamma^{\prime}(p)}{\Gamma(p)}\quad
z,\,p\in\mathds{C}^{+}.$ (84)

  2. (F2)

In Proposition 3, we used the $\beta$-function which was originally derived
from the digamma function via the formula

$\beta(x):=\frac{1}{2}\,\left[\psi\left(\frac{x+1}{2}\right)-\psi\left(\frac{x}{2}\right)\right]=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{x+k},\quad-x\notin\mathbb{N}\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.372(1)})}.$

  3. (F3)

To formulate Proposition 4, we need another special function, namely, _the
exponential integral function_ ${\rm Ei}$, defined by

${\rm Ei}(x):=-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt,\quad
x<0\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.211(1)})}.$ (85)

For $x>0$, ${\rm Ei}$ is defined by _the Cauchy principal value_ (P.V.)
method:

${\rm Ei}(x)=-\lim_{\epsilon\to
0}\left[\int_{-x}^{-\epsilon}\frac{e^{-t}}{t}dt+\int_{\epsilon}^{\infty}\frac{e^{-t}}{t}dt\right],\quad
x>0\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.211(2)})}.$

Other useful representation is

${\rm Ei}(x)=\gamma+\log|x|+\sum_{k=1}^{\infty}\frac{x^{k}}{k\,.\,k!},\quad
x\neq 0\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.214(1,2)})}.$

From above, we get:

$\frac{d}{dx}{\rm Ei}(\pm x)=\frac{e^{\pm x}}{x}\quad\mbox{and}\quad\lim_{x\to
0^{+}}\,{\rm Ei}(\pm
x)-\log|x|=\gamma=\int_{0}^{1}\frac{1-e^{-t}}{t}dt-\int_{1}^{\infty}\frac{e^{-t}}{t}dt.$
(86)

We also have

$\frac{1}{2}\,\left[e^{-x}{\rm Ei}(x)+e^{x}{\rm
Ei}(-x)\right]=-\int_{0}^{\infty}\frac{t\cos(xt)}{1+t^{2}}\,dt,\quad
x\in\mathds{R}\qquad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{grad}{\@@citephrase{,
}}{}, {8.217(2)})}$ (87)

  4. (F4)

The identity

$i\log\frac{1-ie^{x}}{1+ie^{x}}=2\arctan(e^{x}),\quad x\in\mathds{R},$ (88)

is easily checkable by differentiation.

  5. (F5)

For Proposition 5, we need another two special functions, namely, the
Riemann’s zeta function, $\zeta$, and the polylogarithm functions ${\rm
Li}_{n}$, defined as follows:

$\displaystyle\zeta(s,a)$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{\infty}\frac{1}{(k+a)^{s}},\quad\Re(s)>1,\;-a\not\in\mathbb{N};$
(89) $\displaystyle{\rm Li}_{s}(z)$ $\displaystyle:=$
$\displaystyle\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}},\quad|z|<1,\;s\in\mathds{C},$
(90)

with the properties, ${\rm Li}_{s}(1)=\zeta(s,1)$, and the representations

$\displaystyle\zeta(s,a)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-ax}\frac{x^{s-1}}{1-e^{-x}}dx,\quad\Re(a)>0,$
(91) $\displaystyle{\rm Li}_{s}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{\Gamma(s)}\int_{1}^{\infty}\frac{\log^{s-1}(x)}{x(x-z)}dx,\quad
z\in\mathds{C}{\footnotesize\setminus}[1,\infty).\quad\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{Lewin}{\@@citephrase{,
}}{}, (7.188),\; p. \;236)},$ (92)

where (91) is clearly a consequence of (84). The name of the polylogarithm
functions is due to the fact that ${\rm Li}_{n},\,n=1,\,2,\dots,$ can be
defined recursively by

${\rm Li}_{n}(z)=\int_{0}^{z}\frac{{\rm Li}_{n-1}(y)}{y}\,dy,\quad{\rm
Li}_{0}(z)=\frac{z}{1-z},\quad{\rm Li}_{1}(z)=-\log(1-z),\quad z<1.$

By (88) and (Lewin, , Formulae (2.3) p. 38 and (4.29) p. 106), we have

${\rm Li}_{2}(ie^{s})-{\rm
Li}_{2}(-ie^{s})+s\log\frac{1-ie^{s}}{1+ie^{s}}=-2i\int_{0}^{s}\frac{x}{\cosh(x)}dx,\quad
s\in\mathds{R},$ (93)

where $K=({\rm Li}_{2}(i)-{\rm Li}_{2}(-i))/2i$ stands for the Catalan
constant $\approx$ 0.9159. In particular, we have:

${\rm Li}_{2}(-ie^{s})+{\rm
Li}_{2}(ie^{s})=2i\sum_{k=1}^{\infty}(-1)^{k}\frac{e^{(2k-1)s}}{(2k-1)^{2}}.$

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