# Meijer’s $G$-Function and Euler’s Differential Equation Revisited

Fritz Gesztesy  Department of Mathematics, Baylor University, Sid Richardson
Bldg., 1410 S. 4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>https://math.artsandsciences.baylor.edu/person/fritz-gesztesy-phd and  Markus
Hunziker  Department of Mathematics, Baylor University, Sid Richardson Bldg.,
1410 S. 4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>https://math.artsandsciences.baylor.edu/person/markus-hunziker-phd

###### Abstract.

We consider the generalized eigenvalue problem for the classical Euler
differential equation and demonstrate its intimate connection with Meijer’s
$G$-functions.

In the course of deriving the solution of the generalized Euler eigenvalue
equation we review some of the basics of generalized hypergeometric functions
and Meijer’s $G$-functions and some of its special cases where the underlying
Mellin-type integrand exhibits higher-order poles.

###### Key words and phrases:

Euler’s differential equation, Meijer’s $G$-function, generalized
hypergeometric functions, strongly singular coefficients.

###### 2010 Mathematics Subject Classification:

Primary: 34A05, 34B30, 34A30, 34M03; Secondary: 34A30, 34E05, 34L10.

###### Contents

  1. 1 Introduction
  2. 2 Basics on Generalized Hypergeometric Functions and Meijer’s $G$-Functions
  3. 3 Meijer’s $G$-Functions in the Presence of Higher-Order Poles
  4. 4 The Generalized Eigenvalue Problem for Euler’s Differential Equation
  5. 5 Some Remarks on the Nonhomogeneous Euler Differential Equation

##  1\. Introduction

While Euler’s $N$th-order homogeneous differential equation,

$\Bigg{(}z^{-N}\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{)}y(z)=0,\quad\lambda_{j}\in{\mathbb{C}},\,1\leq
j\leq N,\;z\in{\mathbb{C}}\backslash\\{0\\},$ (1.1)

and its nonhomogeneous version,

$\Bigg{(}z^{-N}\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{)}y(z)=f(z),\quad\lambda_{j}\in{\mathbb{C}},\,1\leq
j\leq N,\;z\in{\mathbb{C}}\backslash\\{0\\},$ (1.2)

has received enormous attention in the ODE literature, see, for instance, [4,
p. 122–123, 131, ], [8, p. 85], [9, p.343–344], [10, p. 141–143, 202], [21, p.
26–27, 53–54], [25, p. 116–117], [27, p. 208–209], and our observations in
Section 5, the associated generalized eigenvalue differential equation,

$\Bigg{(}z^{-N}\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{)}y(\mu,z)=\mu
y(\mu,z),\quad\lambda_{j}\in{\mathbb{C}},\,1\leq j\leq
N,\;z\in{\mathbb{C}}\backslash\\{0\\},$ (1.3)

with $\mu\in{\mathbb{C}}$ the spectral parameter, seems to have escaped
scrutiny, at least, we were not able to find pertinent references.

This paper aims at filling this gap and demonstrate that the solutions of
(1.3) can be expressed in terms of Meijer’s $G$-function, and, generically in
terms of generalized hypergeometric functions (see Theorems 4.1 and 4.3).

In Section 2 we recall some of the basic facts on generalized hypergeometric
functions and Meijer’s $G$-functions, relevant in our context and, in the
generic case of the parameters involved (see (2.5) and (2.49), describe a
fundamental system of solutions for the generalized hypergeometric
differential equation (2.4) and a fundamental system of the Meijer
$G$-function differential equation (2.26).

The case of fundamental systems in the general (non-generic) case for Meijer’s
$G$-function $G^{m,n}_{p,q}$ is incredibly complex and beyond the scope of
this note. Instead, in Section 3, we only discuss the special situation
$G^{m,0}_{0,q}$, a case that is tailored precisely to the generalized Euler
eigenvalue problem at hand. In particular, we permit higher order poles in the
integrand in the Mellin-type integral for $G^{m,0}_{0,q}$. We have not been
able to locate this result, even in the present special case considered, in
the literature.

The results of Section 3 are then applied in our principal Section 4 to derive
the solutions of (1.3).

Finally, in Section 5 we add some remarks on the nonhomogeneous $N$th-order
Euler differential equation (1.2).

##  2\. Basics on Generalized Hypergeometric Functions and Meijer’s
$G$-Functions

In this preparatory section we recall some of the basic properties of
generalized hypergeometric functions and Meijer’s $G$-functions. For details
on generalized hypergeometric functions and Meijer’s $G$-functions we refer,
for instance, to [3], [6, Ch. IV, Sects. 5.3–5.6], [11, Ch. V], [13, Ch. V],
[14, Chs. I, II, IV], [19, Ch. 16], and [22, Sect. 8.2].

We start with generalized hypergeometric functions. Introducing111We employ
the convenient abbreviation ${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$.

$\displaystyle\begin{split}&{}_{r}F_{s}\left(\\!\begin{array}[]{c}{\displaystyle\alpha_{1},\dots,\alpha_{r}}\\\
{\displaystyle\beta_{1},\dots,\beta_{s}}\end{array}\bigg{|}\,\zeta\right)=\sum_{k\in{\mathbb{N}}_{0}}\frac{\prod_{\ell=1}^{r}(\alpha_{\ell})_{k}}{\prod_{m=1}^{s}(\beta_{m})_{k}}\,\frac{\zeta^{k}}{k!},\quad\zeta\in{\mathbb{C}},\\\
&r,s\in{\mathbb{N}}_{0},\;r\leq s,\;\alpha\in{\mathbb{C}},\,1\leq\ell\leq
r,\;\beta_{m}\in{\mathbb{C}}\backslash\\{-{\mathbb{N}}_{0}\\},\,1\leq m\leq
s,\end{split}$ (2.1)

with

$(\gamma)_{0}=1,\;(\gamma)_{k}=\gamma(\gamma+1)\cdots(\gamma+k-1)=\Gamma(\gamma+k)/\Gamma(\gamma),\;k\in{\mathbb{N}},\;\gamma\in{\mathbb{C}},$
(2.2)

denoting Pochhammer’s symbol.

Then
${}_{r}F_{s}\Big{(}\\!\begin{array}[]{c}{\scriptstyle\alpha_{1},\dots,\alpha_{r}}\\\
{\scriptstyle\beta_{1},\dots,\beta_{s}}\end{array}\Big{|}\,\cdot\Big{)}$,
$r\leq s$, is entire,

${}_{r}F_{s}\left(\\!\begin{array}[]{c}{\displaystyle\alpha_{1},\dots,\alpha_{r}}\\\
{\displaystyle\beta_{1},\dots,\beta_{s}}\end{array}\bigg{|}\,\zeta\right)\underset{\zeta\to
0}{=}1+O(\zeta),$ (2.3)

and it satisfies the differential equation of order $s+1$ (see [6, p. 184],
[11, p. 136–137], [13, p. 190]),

$\Bigg{[}\zeta\frac{d}{d\zeta}\prod_{m=1}^{s}\bigg{(}\zeta\frac{d}{d\zeta}+\beta_{m}-1\bigg{)}-\zeta\prod_{\ell=1}^{r}\bigg{(}\zeta\frac{d}{d\zeta}+\alpha_{\ell}\bigg{)}\Bigg{]}u(\zeta)=0,\quad\zeta\in{\mathbb{C}}\backslash\\{0\\}.$
(2.4)

Since $r\leq s$, $\zeta=0$ is a regular singularity and $\zeta=\infty$ is an
irregular (i.e., essential) singularity of (2.4); there are no other
singularities of (2.4).

If one strengthens the conditions on $\alpha_{\ell}$, $1\leq\ell\leq r$,
$\beta_{m}$, $1\leq m\leq s$, in (2.1) to

$\displaystyle\begin{split}&r,s\in{\mathbb{N}}_{0},\;r\leq
s,\;\alpha_{\ell}\in{\mathbb{C}},\,1\leq\ell\leq
r,\;\beta_{m}\in{\mathbb{C}}\backslash\\{-{\mathbb{N}}_{0}\\},\,1\leq m\leq
s,\\\ &\beta_{m}-\beta_{m^{\prime}}\in{\mathbb{C}}\backslash{\mathbb{Z}},\quad
1\leq m,m^{\prime}\leq s,\;m\neq m^{\prime},\end{split}$ (2.5)

the following $s+1$ functions

$\displaystyle
u_{0}(\zeta)={}_{r}F_{s}\left(\\!\begin{array}[]{c}{\displaystyle\alpha_{1},\dots,\alpha_{r}}\\\
{\displaystyle\beta_{1},\dots,\beta_{s}}\end{array}\bigg{|}\,\zeta\right),$
(2.8) $\displaystyle u_{m}(\zeta)=\zeta^{1-\beta_{m}}$ (2.9)
$\displaystyle\quad\times{}_{r}F_{s}\left(\\!\begin{array}[]{c}{\displaystyle
1+\alpha_{1}-\beta_{m},\dots\dots\dots\dots\dots\dots.\,.\dots\dots\dots\dots\dots\dots,1+\alpha_{r}-\beta_{m}}\\\
{\displaystyle
1+\beta_{1}-\beta_{m},\dots,1+\beta_{m-1}-\beta_{m},1+\beta_{m+1}-\beta_{m},\dots,1+\beta_{s}-\beta_{m}}\end{array}\bigg{|}\,\zeta\right),$
(2.12) $\displaystyle\hskip 284.52756pt1\leq m\leq
s;\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

constitute a fundamental system of solutions of (2.4) (see, [11, p. 137], [13,
p. 190], and [19, p. 409]). In general, the solutions $u_{m}$, $1\leq m\leq
s$, in (2.5) acquire the branch cut inherited from
$\zeta\mapsto\zeta^{1-\beta_{m}}$.

The setup in (2.4)–(2.9) formally corresponds to introducing $\beta_{0}=1$ in
the first term of (2.4).

######  Remark 2.1.

$(i)$ Our notation in (2.9) (and later on) is somewhat defective: For
instance, the case $m=1$ in (2.9) leads to nonexistent terms as the
$\beta$-row would necessarily start with $1+\beta_{2}-\beta_{1}$. But we
prefer this ambiguity over introducing various case distinctions,
unnecessarily complicating the presentation.
$(ii)$ The restriction $\beta_{m}\in{\mathbb{C}}\backslash\\{0\\}$, $1\leq
m\leq s$, in (2.1) is necessitated by the fact that
${}_{r}F_{s}\Big{(}\\!\begin{array}[]{c}{\scriptstyle\alpha_{1},\dots,\alpha_{r}}\\\
{\scriptstyle\beta_{1},\dots,\beta_{s}}\end{array}\Big{|}\,\cdot\Big{)}$
exhibits a first order pole as $\beta_{m}\to-n$, $n\in{\mathbb{N}}_{0}$. This
yields a simple recipe to define a closely related function, ${}_{r}{\bf
F}_{s}\Big{(}\\!\begin{array}[]{c}{\scriptstyle\alpha_{1},\dots,\alpha_{r}}\\\
{\scriptstyle\beta_{1},\dots,\beta_{s}}\end{array}\Big{|}\,\cdot\Big{)}$,
which is well-defined for all $\beta_{m}\in{\mathbb{C}}$, $1\leq m\leq s$,
viz.,

$\displaystyle{}_{r}{\bf
F}_{s}\Big{(}\\!\begin{array}[]{c}{\scriptstyle\alpha_{1},\dots,\alpha_{r}}\\\
{\scriptstyle\beta_{1},\dots,\beta_{s}}\end{array}\Big{|}\,\cdot\Big{)}$
$\displaystyle={}_{r}F_{s}\Big{(}\\!\begin{array}[]{c}{\scriptstyle\alpha_{1},\dots,\alpha_{r}}\\\
{\scriptstyle\beta_{1},\dots,\beta_{s}}\end{array}\Big{|}\,\cdot\Big{)}\bigg{/}\prod_{m=1}^{s}\Gamma(\beta_{m})$
(2.17)
$\displaystyle=\sum_{k\in{\mathbb{N}}_{0}}\frac{\prod_{\ell=1}^{r}(\alpha_{\ell})_{k}}{\prod_{m=1}^{s}\Gamma(\beta_{m}+k)}\,\frac{\zeta^{k}}{k!},\quad\zeta\in{\mathbb{C}},$
(2.18) $\displaystyle r,s\in{\mathbb{N}}_{0},\;r\leq
s,\;\alpha,\,1\leq\ell\leq r,\;\beta_{m}\in{\mathbb{C}},\,1\leq m\leq s,$

a device used, for instance, in [20, Sect. 5.9] and [19, Nos. 15.2.2, 16.2.5].
$\diamond$

For additional background information in connection with generalized
hypergeometric functions we refer, for instance, to the monographs, [1, Ch.
11], [2, Ch. II], [6, Ch. IV], [11, Chs. V–VII], [12, Ch. IX], [13, Ch. V,
VIII], [14, Sect. 2.3], [19, Ch. 16], [22, Sect. 7.2], [24, Ch. 2].

Next, we turn to some of the basics of Meijer’s $G$-functions and hence
introduce

$\displaystyle G_{p,q}^{m,n}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},\dots,a_{n};a_{n+1},\dots,a_{p}}\\\ {\displaystyle
b_{1},\dots,b_{m};b_{m+1},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(2.21) $\displaystyle\quad=\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1},\dots,b_{m}}}d\omega\,\frac{\prod_{j=1}^{m}\Gamma(b_{j}-\omega)\prod_{k=1}^{n}\Gamma(1-a_{k}+\omega)}{\prod_{j=m+1}^{q}\Gamma(1-b_{j}+\omega)\prod_{k=n+1}^{p}\Gamma(a_{k}-\omega)}\,\zeta^{\omega},$
(2.22) $\displaystyle\hskip 85.35826pt0\leq n\leq p,\;0\leq m\leq q,\;q\geq
1,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

assuming that no pole of $\Gamma(b_{j}-\cdot)$, $1\leq j\leq m$, coincides
with a pole of $\Gamma(1-a_{k}+\cdot)$, $1\leq k\leq n$, thus,

$a_{k}-b_{j}\in{\mathbb{C}}\backslash{\mathbb{N}},\quad 1\leq j\leq m,\;1\leq
k\leq n.$ (2.23)

Here

${\mathcal{C}}_{b_{1},\dots,b_{m}}$ is a contour beginning and ending at
$+\infty$, encircling all poles of $\Gamma(b_{j}-\cdot)$, $1\leq j\leq m$,
once in negative orientation, (2.24) but none of the poles of
$\Gamma(1-a_{k}+\cdot)$, $1\leq k\leq n$,

and the left-hand side of (2.22) is defined as the (absolutely convergent) sum
of residues of the right-hand side.

As a result, (2.22) is trivial if $m=0$,

$G_{p,q}^{0,n}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},\dots,a_{n};a_{n+1},\dots,a_{p}}\\\
{\displaystyle;b_{1},.\dots\dots\dots\dots\dots,b_{q}}\end{array}\bigg{|}\,\,\cdot\,\right)=0.$
(2.25)

One verifies that $G_{p,q}^{m,n}\Big{(}\\!\begin{array}[]{c}{\scriptstyle
a_{1},\dots,a_{n};a_{n+1},\dots,a_{p}}\\\ {\scriptstyle
b_{1},\dots,b_{m};b_{m+1},\dots,b_{q}}\end{array}\Big{|}\,\cdot\Big{)}$,
$p<q$, satisfies the differential equation of order $q$ (see [6, p. 210], [11,
p. 181], [13, p. 192], [14, p. 13, 16]),

$\Bigg{[}\prod_{j=1}^{q}\bigg{(}\zeta\frac{d}{d\zeta}-b_{j}\bigg{)}-(-1)^{m+n-p}\zeta\prod_{k=1}^{p}\bigg{(}\zeta\frac{d}{d\zeta}-a_{k}+1\bigg{)}\Bigg{]}u(\zeta)=0,\quad\zeta\in{\mathbb{C}}\backslash\\{0\\}.$
(2.26)

Again, since $p<q$, $\zeta=0$ is a regular singularity and $\zeta=\infty$ is
an irregular (i.e., essential) singularity of (2.26); there are no other
singularities of (2.26).

Next, if

$b_{j}-b_{j^{\prime}}\in{\mathbb{C}}\backslash{\mathbb{Z}},\quad 1\leq
j,j^{\prime}\leq m,\;j\neq j^{\prime},$ (2.27)

then (cf. [6, p. 208], [11, p. 145], [13, p. 171], [14, p. 3–4])

$\displaystyle G_{p,q}^{m,n}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},\dots,a_{n};a_{n+1},\dots,a_{p}}\\\ {\displaystyle
b_{1},\dots,b_{m};b_{m+1},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(2.30)
$\displaystyle\quad=\sum_{j=1}^{m}\frac{\prod_{j^{\prime}=1\,j^{\prime}\neq
j}^{m}\Gamma(b_{j^{\prime}}-b_{j})\prod_{k=1}^{n}\Gamma(1+b_{j}-a_{k})}{\prod_{j^{\prime}=m+1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})\prod_{k=n+1}^{p}\Gamma(a_{k}-b_{j})}\,\zeta^{b_{j}}$
(2.31)
$\displaystyle\qquad\times{}_{p}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle
1+b_{j}-a_{1},\dots\dots\dots\dots\dots\,.\,\,.\,\,.\,\dots\dots\dots\dots\dots,1+b_{j}-a_{p}}\\\
{\displaystyle
1+b_{j}-b_{1},\dots,1+b_{j}-b_{j-1},1+b_{j}-b_{j+1},\dots,1+b_{j}-b_{q}}\end{array}\bigg{|}$
(2.34) $\displaystyle\hskip 304.4444pt\bigg{|}\,(-1)^{p-m-n}\zeta\bigg{)},$
$\displaystyle\hskip 149.37697pt0\leq n\leq p,\;0\leq m\leq q,\;q\geq
1,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\}.$

In particular, if $m=1$ (and hence (2.27) becomes irrelevant), then (cf. [13,
p. 174], [14, p. 4])

$\displaystyle G_{p,q}^{1,n}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},...\dots,a_{n};a_{n+1},\dots...,a_{p}}\\\ {\displaystyle
b_{j};b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(2.37)
$\displaystyle\quad=\frac{\prod_{k=1}^{n}\Gamma(1+b_{j}-a_{k})}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})\prod_{k=n+1}^{p}\Gamma(a_{k}-b_{j})}\,\zeta^{b_{j}}$
(2.38)
$\displaystyle\qquad\times{}_{p}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle
1+b_{j}-a_{1},\dots\dots\dots\dots\dots\,.\,\,.\,\,.\,\dots\dots\dots\dots\dots,1+b_{j}-a_{p}}\\\
{\displaystyle
1+b_{j}-b_{1},\dots,1+b_{j}-b_{j-1},1+b_{j}-b_{j+1},\dots,1+b_{j}-b_{q}}\end{array}\bigg{|}$
(2.41) $\displaystyle\hskip 307.28984pt\bigg{|}\,(-1)^{p-1-n}\zeta\bigg{)},$
$\displaystyle\hskip 186.36563pt0\leq n\leq p,\;1\leq j\leq
q,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

and (still assuming (2.27)),

$\displaystyle G_{p,q}^{m,n}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},\dots,a_{n};a_{n+1},\dots,a_{p}}\\\ {\displaystyle
b_{1},\dots,b_{m};b_{m+1},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(2.44)
$\displaystyle\quad=\sum_{j=1}^{m}\frac{\prod_{j^{\prime}=1\,j^{\prime}\neq
j}^{m}\Gamma(b_{j^{\prime}}-b_{j})\Gamma(1+b_{j}-b_{j^{\prime}})}{\prod_{k=n+1}^{p}\Gamma(1+b_{j}-a_{j^{\prime}})\Gamma(a_{k}-b_{j})}\,e^{-i\pi
b_{j}(p+1-m-n)}$ (2.45) $\displaystyle\qquad\times
G_{p,q}^{1,p}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},..\dots\dots\dots\dots\dots\dots.,a_{p};}\\\ {\displaystyle
b_{j};b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\end{array}\bigg{|}\,(-1)^{p+1-m-n}\zeta\right),$
(2.48) $\displaystyle\hskip 62.59596pt0\leq n\leq p,\;0\leq m\leq q,\;q\geq
1,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\}$

(see, [13, p. 175], [14, p. 14]).

######  Remark 2.2.

Given Remark 2.1 $(ii)$, the presence of the factors
$\prod_{j^{\prime}=m+1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})$, respectively,
$\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})$, in the denominators
of the right-hand sides of (2.31), respectively, (2.38), renders these right-
hand sides well-defined without imposing the additional condition
$b_{j}-b_{j^{\prime}}\in{\mathbb{C}}\backslash{\mathbb{Z}}$, $1\leq j\leq m$,
$m+1\leq j^{\prime}\leq q$. $\diamond$

We also recall that if

$b_{j}-b_{j^{\prime}}\in{\mathbb{C}}\backslash{\mathbb{Z}},\quad 1\leq
j,j^{\prime}\leq q,\;j\neq j^{\prime},$ (2.49)

then (cf. [11, p. 181], [13, p. 190], [22, p. 621–622])

$\displaystyle G_{p,q}^{1,p}\left(\\!\begin{array}[]{c}{\displaystyle
a_{1},..\dots\dots\dots\dots\dots\dots.,a_{p};}\\\ {\displaystyle
b_{j};b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(2.52)
$\displaystyle\quad=\frac{\prod_{k=1}^{p}\Gamma(1+b_{j}-a_{k})}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})}\,\zeta^{b_{j}}$
(2.53)
$\displaystyle\qquad\times{}_{p}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle
1+b_{j}-a_{1},\dots\dots\dots\dots\dots\,.\,\,.\,\,.\,\dots\dots\dots\dots\dots,1+b_{j}-a_{p}}\\\
{\displaystyle
1+b_{j}-b_{1},\dots,1+b_{j}-b_{j-1},1+b_{j}-b_{j+1},\dots,1+b_{j}-b_{q}}\end{array}\bigg{|}$
(2.56) $\displaystyle\hskip 341.43306pt\bigg{|}-\zeta\bigg{)},$
$\displaystyle\hskip 239.00298pt1\leq j\leq
q,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

constitutes a fundamental system of solutions of (2.26), subject to (2.49).
Equivalently, taking into account the asymptotics (2.3),

$\displaystyle\zeta^{b_{j}}{}_{p}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle
1+b_{j}-a_{1},\dots\dots\dots\dots\dots\,.\,\,.\,\,.\,\dots\dots\dots\dots\dots,1+b_{j}-a_{p}}\\\
{\displaystyle
1+b_{j}-b_{1},\dots,1+b_{j}-b_{j-1},1+b_{j}-b_{j+1},\dots,1+b_{j}-b_{q}}\end{array}\bigg{|}-\zeta\bigg{)},$
(2.59) $\displaystyle\hskip 204.85974pt1\leq j\leq
q,\;p<q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$ (2.60)

represents a fundamental system of solutions of (2.26), subject to (2.49)
(see, [11, p. 181]).

The case when condition (2.49) is violated is much more involved and we will
study a special case, particularly suited for our application to Euler’s
differential equation, in the next section.

For additional results on Meijer’s $G$-function see, for instance, [1, Sect.
11.3], [3], [5], [6, Ch. V], [7], [11, Chs. V, VI], [12, Ch. IX], [13, Ch. V,
VIII], [14], [15]–[18], [19, Ch. 16], [22, Sect. 8.2], [23], [24, Ch. 2],
[28], [29].

##  3\. Meijer’s $G$-Functions in the Presence of Higher-Order Poles

In this section we take a close look at a fundamental system of solutions of
then differential equation (2.26) in the absence of condition (2.27)
(surprisingly, we were not able to locate the details in the literature). In
this case the integrand on the right-hand side of (2.22) exhibits higher-order
poles and finding a fundamental set of solutions of (2.26) becomes a rather
complex task.

To analyze the right-hand side of (2.22) in the case the integrand has higher-
order poles we start with some preparations:
${\bf(I)}$ Let $c\in{\mathbb{C}}$. Then $\Gamma(c-\,\cdot\,)$ has the partial
fraction expansion (see, e.g., [26, p. 162])

$\Gamma(c-\omega)=\underbrace{\int_{1}^{\infty}dx\,e^{-x}x^{c-1-\omega}}_{\text{entire
w.r.t. $\omega$}}+\sum_{n\in{\mathbb{N}}_{0}}\frac{(-1)^{n+1}}{n!(\omega-
c-n)},\quad\omega\in{\mathbb{C}}\backslash\\{c+{\mathbb{N}}_{0}\\}.$ (3.1)

${\bf(II)}$ Let $b\in{\mathbb{C}}$. Then $\Gamma(b-\omega)$ has first order
poles precisely at the points $\omega\in b+{\mathbb{N}}_{0}$ and is analytic
otherwise on ${\mathbb{C}}$. The residues of $\Gamma(b-\omega)$ at $\omega\in
b+{\mathbb{N}}_{0}$, are given by $(-1)^{n+1}/\Gamma(n+1)$, that is,

$\lim_{\omega\to b+n}(\omega-b-n)\Gamma(b-\omega)=(-1)^{n+1}/(n!),\quad
n\in{\mathbb{N}}_{0}.$ (3.2)

(We use the usual convention $0!=1$.)
${\bf(III)}$ Let $b\in{\mathbb{C}}$. Then $1/\Gamma(1-b+\omega)$ is entire
with respect to $\omega$ with simple zeros at $\omega\in b-{\mathbb{N}}$;
there are no other zeros in ${\mathbb{C}}$.

Consequently, the poles of $\Gamma(b_{j}-\,\cdot\,)$ and zeros of
$1/\Gamma(1-b_{j^{\prime}}+\,\cdot\,)$, $1\leq j,j^{\prime}\leq q$, cannot
interfere (and possibly cancel each other).
${\bf(IV)}$ The map
${\mathbb{C}}\ni\zeta\mapsto\zeta^{\omega}=e^{\omega\operatorname{ln}(\zeta)}$
has a cut in $\zeta$ (connecting $\zeta=0$ with infinity), but no cut in
$\omega$.
${\bf(V)}$ Let $\Omega\subseteq{\mathbb{C}}$ be open and connected,
$\omega_{0}\in\Omega$, and $f:\Omega\backslash\\{\omega_{0}\\}\to{\mathbb{C}}$
analytic, with $\omega_{0}$ a pole of $f$ of order $k\in{\mathbb{N}}$. Then
the residue of $f$ at $\omega_{0}$ is given by

${\rm
Res\,}_{\omega_{0}}(f(\,\cdot\,))=\lim_{\omega\to\omega_{0}}\frac{1}{(k-1)!}\frac{d^{k-1}}{d\omega^{k-1}}\big{[}(\omega-\omega_{0})^{k}f(\omega)\big{]}.$
(3.3)

${\bf(VI)}$ For subsequent use we also note the elementary fact,

$\frac{d^{k-1}}{d\omega^{k-1}}\zeta^{\omega}=\zeta^{\omega}[\operatorname{ln}(\zeta)]^{k-1},\quad\zeta\in{\mathbb{C}}\backslash\\{0\\},\;k\in{\mathbb{N}}.$
(3.4)

Next, going beyond condition (2.49) we decompose the multiset
$\\{b_{1},\dots,b_{q}\\}$ into congruence classes ${\rm mod}\,{\mathbb{Z}}$ as
follows:

$\displaystyle\begin{split}&\\{b_{1},\dots,b_{q}\\}=\\{\underbrace{b_{1},\dots,b_{n_{1}}}_{\text{$n_{1}$
terms}}\\}\cup\\{\underbrace{b_{n_{1}+1},\dots,b_{n_{1}+n_{2}}}_{\text{$n_{2}$
terms}}\\}\cup\cdots\\\ &\hskip
71.13188pt\cdots\cup\\{\underbrace{b_{n_{1}+\cdots+n_{r-1}+1},\dots,b_{n_{1}+\cdots+n_{r-1}+n_{r}}\equiv
b_{q}}_{\text{$n_{r}$ terms}}\\},\end{split}$ (3.5)
$\displaystyle\sum_{s=1}^{r}n_{s}=q,$ (3.6)
$\displaystyle\begin{split}&\operatorname{Re}(b_{1})\geq\cdots\geq\operatorname{Re}(b_{n_{1}}),\\\
&\operatorname{Re}(b_{n_{1}+1})\geq\cdots\geq\operatorname{Re}(b_{n_{1}+n_{2}}),\\\
&\quad\vdots\\\ &\operatorname{Re}(b_{n_{1}+\cdots
n_{r-1}+1})\geq\cdots\geq\operatorname{Re}(b_{n_{1}+\cdots
n_{r-1}+n_{r}}),\end{split}$ (3.7)
$\displaystyle\begin{split}&b_{j}-b_{j^{\prime}}\in{\mathbb{Z}},\;1\leq
j,j^{\prime}\leq n_{1},\\\
&b_{n_{1}+j}-b_{n_{1}+j^{\prime}}\in{\mathbb{Z}},\;1\leq j,j^{\prime}\leq
n_{2},\\\ &\quad\vdots\\\
&b_{n_{1}+\cdots+n_{r-1}+j}-b_{n_{1}+\cdot+n_{r-1}+j^{\prime}}\in{\mathbb{Z}},\;1\leq
j,j^{\prime}\leq n_{r}.\end{split}$ (3.8)

One notes that

$G_{0,q}^{m,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{1},\dots,b_{m};b_{m+1},\dots,b_{q}}\end{array}\bigg{|}\,(-1)^{m}\zeta\right),\quad
1\leq m\leq q,$ (3.9)

satisfies the ordinary differential equation of order $q$,

$\Bigg{[}\prod_{j=1}^{q}\bigg{(}\zeta\frac{d}{d\zeta}-b_{j}\bigg{)}-\zeta\Bigg{]}u(\zeta)=0,\quad\zeta\in{\mathbb{C}}\backslash\\{0\\}.$
(3.10)

Case (1): In the generic case, where all congruence classes consist of
precisely one element, equivalently, when $r=q$, then linearly independent
solutions, in fact, a fundamental system of solutions of (3.10), is given by

$\displaystyle G_{0,q}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{j};b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\end{array}\bigg{|}-\zeta\right)=\frac{\zeta^{b_{j}}}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{j}-b_{j^{\prime}})}$
(3.13)
$\displaystyle\qquad\times{}_{0}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+b_{j}-b_{1},\dots,1+b_{j}-b_{j-1},1+b_{j}-b_{j+1},\dots,1+b_{j}-b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$
(3.16) $\displaystyle\quad\underset{\zeta\to
0}{=}C_{j}\zeta^{b_{j}}[1+O(\zeta)],\quad
C_{j}\in{\mathbb{C}}\backslash\\{0\\},\;1\leq j\leq
q,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$ (3.17)

a special case of the situation mentioned in Section 3.
Case (2): It suffices to consider the congruence class
$\\{b_{1},\dots,b_{n_{1}}\\}$ in some detail as the remaining ones are treated
in exactly the same manner. The existence of a system of linearly independent
solutions of (3.10) will be demonstrated by the different asymptotic behavior
of solutions as $\zeta\to 0$. The treatment of all congruence classes in (3.5)
then yields a fundamental system of solutions of (3.10).

Referring to (3.9), one obtains the system of solutions,

$\displaystyle y_{1}(\zeta)$
$\displaystyle=G_{0,q}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle b_{1};b_{2},\dots,b_{q}}\end{array}\bigg{|}-\zeta\right)$
(3.20)
$\displaystyle=\frac{\zeta^{b_{1}}}{\prod_{j^{\prime}=1}^{q}\Gamma(1-b_{1}-b_{j^{\prime}})}\,{}_{0}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+b_{1}-b_{2},\dots,1+b_{1}-b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$ (3.23)
$\displaystyle\underset{\zeta\to 0}{=}C_{1,1}\zeta^{b_{1}}[1+O(\zeta)],$
$\displaystyle y_{2}(\zeta)$
$\displaystyle=G_{0,q}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle b_{1},b_{2};b_{3}\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right)$
(3.26) $\displaystyle\underset{\zeta\to
0}{=}C_{2,1}\zeta^{b_{1}}\operatorname{ln}(\zeta)[1+O(\zeta)]+D_{2,1,b_{1}}\zeta^{b_{1}}[1+O(\zeta)]+C_{2,2}\zeta^{b_{2}}[1+O(\zeta)],$
$\displaystyle\underset{\zeta\to
0}{=}C_{2,1}\zeta^{b_{1}}\operatorname{ln}(\zeta)[1+O(\zeta)]+C_{2,2}\zeta^{b_{2}}[1+O(\zeta)],$
$\displaystyle\vdots$ $\displaystyle y_{n_{1}}(\zeta)$
$\displaystyle=G_{0,q}^{n_{1},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{1},\dots,b_{n_{1}};b_{n_{1}+1},\dots,b_{q}}\end{array}\bigg{|}\,(-1)^{n_{1}}\zeta\right)$
(3.29) $\displaystyle\underset{\zeta\to
0}{=}C_{n_{1},1}\zeta^{b_{1}}[\operatorname{ln}(\zeta)]^{n_{1}-1}[1+O(\zeta)]+\zeta^{b_{1}}\sum_{k=2}^{n_{1}}D_{n_{1},k,b_{1}}[\operatorname{ln}(\zeta)]^{n_{1}-k}[1+O(\zeta)]$
$\displaystyle\quad\;+C_{n_{1},2}\zeta^{b_{2}}[\operatorname{ln}(\zeta)]^{n_{1}-2}[1+O(\zeta)]+\zeta^{b_{2}}\sum_{k=3}^{n_{1}}D_{n_{1},k,b_{2}}[\operatorname{ln}(\zeta)]^{n_{1}-k}[1+O(\zeta)]$
$\displaystyle\quad\;+\dots+C_{n_{1},n_{1}}\zeta^{b_{n-1}}[1+O(\zeta)]$
$\displaystyle\underset{\zeta\to
0}{=}C_{n_{1},1}\zeta^{b_{1}}[\operatorname{ln}(\zeta)]^{n_{1}-1}[1+O(\zeta)]+C_{n_{1},2}\zeta^{b_{2}}[\operatorname{ln}(\zeta)]^{n_{1}-2}[1+O(\zeta)]$
$\displaystyle\quad\;+\dots+C_{n_{1},n_{1}}\zeta^{b_{n_{1}}}[1+O(\zeta)];\quad\zeta\in{\mathbb{C}}\backslash\\{0\\},$
(3.30)

where we used the ordering of the $b$’s in (3.7) to ignore all the $D$-terms
corresponding to the coefficients
$D_{2,1,b_{1}},\dots,D_{n_{1},n_{1},b_{n_{1}-1}}$. In addition,

all terms of the type $[1+O(\zeta)]$ are analytic at $\zeta=0$, (3.31)

and

$\displaystyle C_{1,1}\in{\mathbb{C}}\backslash\\{0\\},$ $\displaystyle
C_{2,2}\in{\mathbb{C}}\backslash\\{0\\}\,\text{ if
$\operatorname{Re}(b_{1})>\operatorname{Re}(b_{2})$,}\quad
C_{2,1}\in{\mathbb{C}}\backslash\\{0\\}\,\text{ if
$\operatorname{Re}(b_{1})=\operatorname{Re}(b_{2})$ (i.e., $b_{1}=b_{2}$),}$
$\displaystyle C_{3,3}\in{\mathbb{C}}\backslash\\{0\\}\,\text{ if
$\operatorname{Re}(b_{2})>\operatorname{Re}(b_{3})$,}\quad C_{3,2}\neq
0\,\text{ if
$\operatorname{Re}(b_{1})>\operatorname{Re}(b_{2})=\operatorname{Re}(b_{3})$,}$
$\displaystyle\quad C_{3,1}\in{\mathbb{C}}\backslash\\{0\\}\,\text{ if
$\operatorname{Re}(b_{1})=\operatorname{Re}(b_{2})=\operatorname{Re}(b_{3})$,}$
(3.32) $\displaystyle\quad\vdots$ $\displaystyle
C_{n_{1},n_{1}}\in{\mathbb{C}}\backslash\\{0\\}\,\text{ if
$\operatorname{Re}(b_{n_{1}-1})>\operatorname{Re}(b_{n_{1}})$,}\dots$
$\displaystyle\quad\dots,C_{n_{1},1}\in{\mathbb{C}}\backslash\\{0\\}\,\text{
if
$\operatorname{Re}(b_{1})=\operatorname{Re}(b_{2})=\dots=\operatorname{Re}(b_{n_{1}})$.}$

Clearly, the smallest $\operatorname{Re}(b_{\ell})$ on the right-hand sides of
(3.30), $1\leq\ell\leq n_{1}$, together with the largest power of
$\operatorname{ln}(\zeta)$ gives rise to the leading asymptotics as $\zeta\to
0$ in each of the solutions $y_{j}$, $1\leq j\leq n_{1}$, in (3.30).

Taking (3) for granted, the asymptotic behavior as $\zeta\to 0$ in (3.30)
shows that the latter represents a system of linearly independent solutions of
(3.10).

To verify the statements in (3.31) and (3), one relies on items
${\bf(I)}$–${\bf(VI)}$ as follows: Starting with the solution $y_{1}$, one
analyzes

$\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1}}}d\omega\,\frac{\Gamma(b_{1}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}(-\zeta)^{\omega},$
(3.33)

where the integrand has simple poles precisely at $\omega\in
b_{1}+{\mathbb{N}}_{0}$ and is analytic in the region enclosed by
${\mathcal{C}}_{b_{1}}$ and in a neighborhood of
$\partial{\mathcal{C}}_{b_{1}}$. Thus, one obtains from the residue theorem
(taking into account that ${\mathcal{C}}_{b_{1}}$ is negatively oriented)

$\displaystyle\begin{split}&\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1}}}d\omega\,\frac{\Gamma(b_{1}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}(-\zeta)^{\omega}\\\
&\quad=-\sum_{n\in{\mathbb{N}}_{0}}{\rm
Res\,}_{\omega=b_{1}+n}\bigg{(}\frac{\Gamma(b_{1}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}(-\zeta)^{\omega}\bigg{)}\\\
&\quad=-\sum_{n\in{\mathbb{N}}_{0}}\lim_{\omega\to
b_{1}+n}\bigg{[}\frac{(\omega-
b_{1}-n)\Gamma(b_{1}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}(-\zeta)^{\omega}\bigg{]}\\\
&\quad=-\sum_{n\in{\mathbb{N}}_{0}}\frac{(-1)^{n+1}}{\Gamma(n+1)}\frac{1}{\prod_{j=2}^{q}\Gamma(1-b_{j}+b_{1}+n)}(-\zeta)^{b_{1}+n}\\\
&\hskip 7.11317pt\underset{\zeta\to
0}{=}\frac{(-1)^{b_{1}}}{\prod_{j=2}^{q}\Gamma(1+b_{1}-b_{j})}\zeta^{b_{1}}\underbrace{[1+O(\zeta)]}_{\text{analytic
at $\zeta=0$}}+D\zeta^{b_{1}+1}\underbrace{[1+O(\zeta)]}_{\text{analytic at
$\zeta=0$}}\\\ &\hskip 7.11317pt\underset{\zeta\to
0}{=}C_{1,1}\zeta^{b_{1}}[1+O(\zeta)],\quad
C_{1,1}\in{\mathbb{C}}\backslash\\{0\\},\;D\in{\mathbb{C}}.\end{split}$ (3.34)

The fact that the $n=0$ term under the sum in (3.34) yields the leading
asymptotic contribution as $\zeta\to 0$ is a general phenomenon that persists
for the remaining solutions $y_{2},\dots,y_{n_{1}}$, in fact, it applies to
all solutions $y_{1},\dots,y_{q}$ of (3.10).

Turning to the solution $y_{2}$, one now considers

$\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1},b_{2}}}d\omega\,\frac{\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega},$
(3.35)

where the integrand now has poles at $\omega\in b_{1}+{\mathbb{N}}_{0}$ and
$\omega\in b_{2}+{\mathbb{N}}_{0}$. More precisely, since
$\operatorname{Re}(b_{1})\geq\operatorname{Re}(b_{2})$, one has
$(b_{1}-b_{2})\in{\mathbb{N}}_{0}$, and thus $b_{1}$ and $b_{2}$ lie on a
straight line parallel to the $\operatorname{Im}(\omega)=0$-axis. In addition,
the integrand in (3.35) has double (and simple) poles at $\omega\in
b_{1}+{\mathbb{N}}_{0}$, and simple poles at
$\\{b_{2},b_{2}+1,\dots,b_{2}+(b_{1}-b_{2})-1\\}$, unless
$\operatorname{Re}(b_{1})=\operatorname{Re}(b_{2})$ (and hence $b_{1}=b_{2}$)
in which case the simple poles are absent.

For the simple poles (assuming temporarily that
$\operatorname{Re}(b_{1})>\operatorname{Re}(b_{2})$) one closely follows the
computation in (3.34), and hence infers

$\displaystyle\begin{split}&\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1},b_{2}}}d\omega\,\frac{\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\\\
&\quad=-\sum_{n=0}^{b_{1}-b_{2}-1}{\rm
Res\,}_{\omega=b_{2}+n}\bigg{(}\frac{\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=3}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{)}\\\
&\quad=-\sum_{n=0}^{b_{1}-b_{2}-1}\lim_{\omega\to
b_{2}+n}\bigg{[}\frac{[(\omega-
b_{2}-n)\Gamma(b_{2}-\omega)]\Gamma(b_{1}-\omega)}{\prod_{j=3}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{]}\\\
&\hskip 7.11317pt\underset{\zeta\to
0}{=}\frac{\Gamma(b_{1}-b_{2})}{\prod_{j=3}^{q}\Gamma(1+b_{1}-b_{j})}\zeta^{b_{2}}\underbrace{[1+O(\zeta)]}_{\text{analytic
at $\zeta=0$}}+D\zeta^{b_{2}+1}\underbrace{[1+O(\zeta)]}_{\text{analytic at
$\zeta=0$}}\\\ &\hskip 7.11317pt\underset{\zeta\to
0}{=}C_{2,2}\zeta^{b_{2}}[1+O(\zeta)],\quad
C_{2,2}\in{\mathbb{C}}\backslash\\{0\\},\;D\in{\mathbb{C}}.\end{split}$ (3.36)

Once again, the leading order contribution as $\zeta\to 0$ stems from the
$n=0$ term under the sum in (3.36).

For the second order poles at $\omega\in b_{1}+{\mathbb{N}}_{0}$, one now
obtains

$\displaystyle\frac{1}{2\pi
i}\int_{{\mathcal{C}}_{b_{1},b_{2}}}d\omega\,\frac{\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=2}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}$
$\displaystyle\quad=-\sum_{n\in{\mathbb{N}}_{0}}{\rm
Res\,}_{\omega=b_{1}+n}\bigg{(}\frac{\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=3}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{)}$
$\displaystyle\quad=-\sum_{n\in{\mathbb{N}}_{0}}\lim_{\omega\to
b_{1}+n}\frac{d}{d\omega}\bigg{[}\frac{(\omega-
b_{1}-n)^{2}\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=3}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{]}$
$\displaystyle\quad=-\lim_{\omega\to
b_{1}}\frac{d}{d\omega}\bigg{[}\frac{(\omega-
b_{1})^{2}\Gamma(b_{1}-\omega)\Gamma(b_{2}-\omega)}{\prod_{j=3}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{]}$
(3.37) $\displaystyle\hskip 7.11317pt\underset{\zeta\to
0}{=}\frac{(-1)^{1+b_{1}-b_{2}}}{\Gamma(1+b_{1}-b_{2})\prod_{j=3}^{q}\Gamma(1+b_{1}-b_{j})}\zeta^{b_{1}}\operatorname{ln}(\zeta)\underbrace{[1+O(\zeta)]}_{\text{analytic
at $\zeta=0$}}$ $\displaystyle\hskip
17.07164pt\quad+D_{2,1,b_{1}}\zeta^{b_{1}}\underbrace{[1+O(\zeta)]}_{\text{analytic
at $\zeta=0$}}$ $\displaystyle\hskip 7.11317pt\underset{\zeta\to
0}{=}C_{2,1}\zeta^{b_{1}}\operatorname{ln}(\zeta)[1+O(\zeta)]+D_{2,1,b_{1}}\zeta^{b_{1}}[1+O(\zeta)],\quad
C_{2,1}\in{\mathbb{C}}\backslash\\{0\\},\;D_{2,1,b_{1}}\in{\mathbb{C}}.$

(We recall that $(b_{1}-b_{2})\in{\mathbb{N}}_{0}$.) Since
$\operatorname{Re}(b_{1})\geq\operatorname{Re}(b_{2})$, the term
$D_{2,1,b_{1}}\zeta^{b_{1}}[1+O(\zeta)]$ is subordinate to
$C_{2,2}\zeta^{b_{2}}[1+O(\zeta)]$ and hence omitted in $y_{2}$ in (3.30).

One can now iterate this procedure for all $y_{3},\dots,y_{n_{1}}$. In
particular, if

$\operatorname{Re}(b_{\ell-1})>\operatorname{Re}(b_{\ell})=\dots=\operatorname{Re}(b_{n_{1}}),\quad
1\leq\ell\leq n_{1},$ (3.38)

then for $n\in{\mathbb{N}}_{0}$, one obtains in connection with the solution
$y_{n_{1}}$

$\displaystyle\begin{split}&\lim_{\omega\to
b_{\ell}+n}\frac{d^{n_{1}-\ell}}{d\omega^{n_{1}-\ell}}\bigg{[}(\omega-
b_{\ell}-n)^{n_{1}-\ell+1}\\\
&\qquad\;\,\times\frac{\Gamma(b_{1}-\omega)\dots\Gamma(b_{\ell-1}-\omega)[\Gamma(b_{\ell}-\omega)\dots\Gamma(b_{n_{1}}-\omega)]}{\prod_{j=n_{1}+1}^{q}\Gamma(1-b_{j}+\omega)}\zeta^{\omega}\bigg{]}\\\
&\hskip 7.11317pt\underset{\zeta\to
0}{=}C_{n_{1},\ell}\zeta^{b_{\ell}}[\operatorname{ln}(\zeta)]^{n_{1}-\ell}[1+O(\zeta)]+\zeta^{b_{\ell}}\sum_{k=\ell+1}^{n_{1}}D_{n_{1},k,b_{\ell}}[\operatorname{ln}(\zeta)]^{n_{1}-k}[1+O(\zeta)],\\\
&\hskip
144.54004ptC_{n_{1},\ell}\in{\mathbb{C}}\backslash\\{0\\},\;D_{n_{1},k,b_{\ell}}\in{\mathbb{C}},\;\ell+1\leq
k\leq n_{1}.\end{split}$ (3.39)

At this point one repeats the analysis for each of the remaining congruence
classes in (3.5). In particular, repeating the analog of (3.30) for the
remaining congruence classes in (3.5), and noting that solutions belonging to
different congruence classes are automatically linearly independent, then
yields a fundamental system of solutions of (3.10).

Summarizing the discussion in this section one arrives at the following
result:

######  Theorem 3.1.

Assume the notation established in (3.5)–(3.10). Then a fundamental system of
solutions of the $q$th-order differential equation (3.10) is given as follows:

$\displaystyle y_{1}(\zeta)$
$\displaystyle=G_{0,q}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle b_{1};b_{2},\dots,b_{q}}\end{array}\bigg{|}-\zeta\right)$
(3.42)
$\displaystyle=\frac{\zeta^{b_{1}}}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{1}-b_{j^{\prime}})}\,{}_{0}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+b_{1}-b_{2},\dots,1+b_{1}-b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$ (3.45)
$\displaystyle y_{2}(\zeta)$
$\displaystyle=G_{0,q}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{1},b_{2};b_{3}\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right),$ (3.48)
$\displaystyle\vdots$ $\displaystyle y_{n_{1}}(\zeta)$
$\displaystyle=G_{0,q}^{n_{1},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{1},\dots,b_{n_{1}};b_{n_{1}+1},\dots,b_{q}}\end{array}\bigg{|}\,(-1)^{n_{1}}\zeta\right),$
(3.51) $\displaystyle y_{n_{1}+1}(\zeta)$
$\displaystyle=G_{0,q}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+1};b_{1},\dots,b_{n_{1}},b_{n_{1}+2},\dots,b_{q}}\end{array}\bigg{|}-\zeta\right)$
(3.54)
$\displaystyle=\frac{\zeta^{b_{n_{1}+1}}}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{n_{1}+1}-b_{j^{\prime}})}$
$\displaystyle\quad\times{}_{0}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+b_{n_{1}+1}-b_{2},\dots,1+b_{n_{1}+1}-b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$
(3.57) $\displaystyle y_{n_{1}+2}(\zeta)$
$\displaystyle=G_{0,q}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+1},b_{n_{1}+2};b_{1},\dots,b_{n_{1}},b_{n_{1}+3},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\right),$
(3.60) $\displaystyle\vdots$ (3.61) $\displaystyle y_{n_{1}+n_{2}}(\zeta)$
$\displaystyle=G_{0,q}^{n_{2},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+1},b_{n_{1}+2},\dots,b_{n_{1}+n_{2}};b_{n_{1}+n_{2}+1},\dots,b_{q}}\end{array}\bigg{|}\,(-1)^{n_{2}}\zeta\right),$
(3.64) $\displaystyle\vdots$ $\displaystyle
y_{n_{1}+n_{2}+\cdots+n_{r-1}+1}(\zeta)$
$\displaystyle=G_{0,q}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+\cdots+n_{r-1}+1};b_{1},\dots,b_{n_{1}+\cdots+n_{r-1}},b_{n_{1}+\cdots+n_{r-1}+2},\dots,b_{q}}\end{array}\bigg{|}\,-\zeta\right)$
(3.67)
$\displaystyle=\frac{\zeta^{b_{n_{1}+\cdots+n_{r-1}+1}}}{\prod_{j^{\prime}=1}^{q}\Gamma(1+b_{n_{1}+\cdots+n_{r-1}+1}-b_{j^{\prime}})}$
$\displaystyle\times{}_{0}F_{q-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+b_{n_{1}+\cdots+n_{r-1}+1}-b_{2},\dots,1+b_{n_{1}+\cdots+n_{r-1}+1}-b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$
(3.70) $\displaystyle y_{n_{1}+n_{2}+\cdots+n_{r-1}+2}(\zeta)$
$\displaystyle=G_{0,q}^{2,0}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+\cdots+n_{r-1}+1},b_{n_{1}+\cdots+n_{r-1}+2};b_{1},\dots,b_{n_{1}+\cdots+n_{r-1}},}\end{array}$
(3.73) $\displaystyle\hskip 88.2037pt\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+\cdots+n_{r-1}+3},\dots,b_{q}}\end{array}\bigg{|}\,\zeta\bigg{)},$
(3.76) $\displaystyle\vdots$ $\displaystyle
y_{n_{1}+n_{2}+\cdots+n_{r-1}+n_{r}}(\zeta)\equiv y_{q}(\zeta)$
$\displaystyle=G_{0,q}^{n_{r},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
b_{n_{1}+\cdots+n_{r-1}+1},\dots,b_{n_{1}+\cdots+n_{r-1}+n_{r}}\equiv
b_{q};b_{1},\dots,b_{n_{1}+\cdots+n_{r-1}}}\end{array}\bigg{|}\,(-1)^{n_{r}}\zeta\right);$
(3.79) $\displaystyle\hskip 294.48619pt\zeta\in{\mathbb{C}}\backslash\\{0\\}.$

##  4\. The Generalized Eigenvalue Problem for Euler’s Differential Equation

To fix our notation we denote by $\tau_{N}(\lambda_{1},\dots,\lambda_{N})$ the
$N$th-order Euler differential expression in the form

$\tau_{N}(\lambda_{1},\dots,\lambda_{N})=z^{-N}\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]},\quad\lambda_{j}\in{\mathbb{C}},\,1\leq
j\leq N,\;z\in{\mathbb{C}}\backslash\\{0\\}$ (4.1)

(noting the commutativity of the factors $[z(d/dz)-\lambda_{j}]$, $1\leq j\leq
N$, on the right-hand side of (4.1)), in the complex domain, and in the
following focus on the generalized eigenvalue problem associated with (4.1),
that is,

$\tau_{N}(\lambda_{1},\dots,\lambda_{N})y(\mu,z)=\mu
y(\mu,z),\quad\lambda_{j}\in{\mathbb{C}},\,1\leq j\leq
N,\;\mu\in{\mathbb{C}},\;z\in{\mathbb{C}}\backslash\\{0\\},$ (4.2)

with $\mu\in{\mathbb{C}}$ the spectral parameter.

To make the connection of (4.2) with Meijer’s $G$-functions we start by noting
the following fact:

######  Theorem 4.1.

Consider the generalized eigenvalue problem for the $N$th-order Euler
differential equation (4.2). Upon introducing the change of variables

$\zeta=\mu(z/N)^{N},\quad\eta(\zeta)=y(\mu,z),$ (4.3)

the generalized eigenvalue problem (4.2) turns into

$\Bigg{[}\prod_{j=1}^{N}\bigg{(}\zeta\frac{d}{d\zeta}-\frac{\lambda_{j}}{N}\bigg{)}-\zeta\Bigg{]}\eta(\zeta)=0,\quad\zeta\in{\mathbb{C}}\backslash\\{0\\},$
(4.4)

the differential equation for
$G_{0,N}^{m,0}\Big{(}\\!\begin{array}[]{c}{\scriptstyle}\hfil\\\
{\scriptstyle\lambda_{1}/N,\dots,\lambda_{m}/N;\lambda_{m+1},\dots,\lambda_{N}/N}\end{array}\Big{|}\,(-1)^{m}\zeta\Big{)}$,
$1\leq m\leq N$, $\zeta\in{\mathbb{C}}\backslash\\{0\\}$.

In particular, assuming the generic case where

$[\lambda_{j}-\lambda_{j^{\prime}}]/N\in{\mathbb{C}}\backslash{\mathbb{Z}},\quad
1\leq j,j^{\prime}\leq N,\;j\neq j^{\prime},$ (4.5)

then

$\displaystyle\eta_{j}(\zeta)$
$\displaystyle=G_{0,N}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\lambda_{j}/N;\lambda_{1}/N,\dots,\lambda_{j-1}/N,\lambda_{j+1}/N,\dots,\lambda_{N}/N}\end{array}\bigg{|}-\zeta\right)$
(4.8)
$\displaystyle=\frac{\zeta^{\lambda_{j}/N}}{\prod_{j^{\prime}=1}^{N}\Gamma\Big{(}1+\frac{\lambda_{j}-\lambda_{j^{\prime}}}{N}\Big{)}}$
(4.9)
$\displaystyle\quad\times{}_{0}F_{N-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+[(\lambda_{j}-\lambda_{1})/N],\dots,1+[(\lambda_{j}-\lambda_{j-1})/N],}\end{array}$
(4.12) $\displaystyle\hskip
68.28644pt\begin{array}[]{c}{\displaystyle}\hfil\\\ {\displaystyle
1+[(\lambda_{j}-\lambda_{j+1})/N],\dots,1+[(\lambda_{j}-\lambda_{N})/N]}\end{array}\bigg{|}\,\zeta\bigg{)},$
(4.15) $\displaystyle\hskip 197.74655pt1\leq j\leq
N,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

represents a fundamental system of solutions of (4.4), subject to (4.5).

Finally,

$\displaystyle y_{j}(\mu,z)$
$\displaystyle=G_{0,N}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\lambda_{j}/N;\lambda_{1}/N,\dots,\lambda_{j-1}/N,\lambda_{j+1}/N,\dots,\lambda_{N}/N}\end{array}\bigg{|}-\mu(z/N)^{N}\right)$
(4.18)
$\displaystyle=\frac{\mu^{\lambda_{j}/N}(z/N)^{\lambda_{j}}}{\prod_{j^{\prime}=1}^{N}\Gamma\Big{(}1+\frac{\lambda_{j}-\lambda_{j^{\prime}}}{N}\Big{)}}$
(4.19)
$\displaystyle\quad\times{}_{0}F_{N-1}\bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+[(\lambda_{j}-\lambda_{1})/N],\dots,1+[(\lambda_{j}-\lambda_{j-1})/N],}\end{array}$
(4.22) $\displaystyle\hskip
72.55461pt\begin{array}[]{c}{\displaystyle}\hfil\\\ {\displaystyle
1+[(\lambda_{j}-\lambda_{j+1})/N],\dots,1+[(\lambda_{j}-\lambda_{N})/N]}\end{array}\bigg{|}\,\mu(z/N)^{N}\bigg{)},$
(4.25) $\displaystyle\hskip 239.00298pt1\leq j\leq
N,\;\zeta\in{\mathbb{C}}\backslash\\{0\\},$

represents a fundamental system of solutions of the generalized Euler
eigenvalue problem (4.2), subject to (4.5).

###### Proof.

While (4.2)–(4.4) follow from elementary computations, (4.5)–(4.9) are a
consequence of (2.53), (2.60). Finally, (4.19) is clear from (4.3) and (4.9).
∎

######  Example 4.2.

Let $N=1$, $\lambda_{1}\in{\mathbb{C}}$, then

$\tau_{1}(\lambda_{1})=z^{-1}\bigg{[}z\frac{d}{dz}-\lambda_{1}\bigg{]},\quad
z\in{\mathbb{C}}\backslash\\{0\\},$ (4.26)

and the solution of

$\tau_{1}(\lambda_{1})y(\mu,z)=\mu
y(\mu,z),\quad\mu\in{\mathbb{C}},\;z\in{\mathbb{C}}\backslash\\{0\\},$ (4.27)

is given by

$y_{1}(\mu,z)=G_{0,1}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\lambda_{1}}\end{array}\bigg{|}-\mu z\right)=(-\mu
z)^{\lambda_{1}}{}_{0}F_{0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle}\hfil\end{array}\bigg{|}\,\mu z\right)=(-\mu
z)^{\lambda_{1}}e^{\mu z},\quad z\in{\mathbb{C}}.$ (4.28)

To treat the non-generic cases where condition (4.5) is violated, we now adapt
the discussion in Section 3 to the present situation. We start by introducing
some notation and decompose the set $\\{\lambda_{1}/N,\dots,\lambda_{N}/N\\}$
into congruence classes ${\rm mod}\,{\mathbb{Z}}$ as follows:

$\displaystyle\begin{split}&\\{\lambda_{1}/N,\dots,\lambda_{N}/N\\}=\\{\underbrace{\lambda_{1}/N,\dots,\lambda_{n_{1}}/N}_{\text{$n_{1}$
terms}}\\}\cup\\{\underbrace{\lambda_{n_{1}+1}/N,\dots,\lambda_{n_{1}+n_{2}}/N}_{\text{$n_{2}$
terms}}\\}\cup\cdots\\\
&\quad\cdots\cup\\{\underbrace{\lambda_{n_{1}+\cdots+n_{r-1}+1}/N,\dots,\lambda_{n_{1}+\cdots+n_{r-1}+n_{r}}/N\equiv\lambda_{N}/N}_{\text{$n_{r}$
terms}}\\},\end{split}$ (4.29) $\displaystyle\sum_{s=1}^{r}n_{s}=N,$ (4.30)
$\displaystyle\begin{split}&\operatorname{Re}(\lambda_{1})\geq\cdots\geq\operatorname{Re}(\lambda_{n_{1}}),\\\
&\operatorname{Re}(\lambda_{n_{1}+1})\geq\cdots\geq\operatorname{Re}(\lambda_{n_{1}+n_{2}}),\\\
&\quad\vdots\\\ &\operatorname{Re}(\lambda_{n_{1}+\cdots
n_{r-1}+1})\geq\cdots\geq\operatorname{Re}(\lambda_{n_{1}+\cdots
n_{r-1}+n_{r}}),\end{split}$ (4.31)
$\displaystyle\begin{split}&[\lambda_{j}-\lambda_{j^{\prime}}]/N\in{\mathbb{Z}},\;1\leq
j,j^{\prime}\leq n_{1},\\\
&[\lambda_{n_{1}+j}-\lambda_{n_{1}+j^{\prime}}]/N\in{\mathbb{Z}},\;1\leq
j,j^{\prime}\leq n_{2},\\\ &\quad\vdots\\\
&[\lambda_{n_{1}+\cdots+n_{r-1}+j}-\lambda_{n_{1}+\cdot+n_{r-1}+j^{\prime}}]/N\in{\mathbb{Z}},\;1\leq
j,j^{\prime}\leq n_{r}.\end{split}$ (4.32)

We also recall that

$G_{0,N}^{m,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\lambda_{1}/N,\dots,\lambda_{m}/N;\lambda_{m+1}/N,\dots,\lambda_{N}/N}\end{array}\bigg{|}\,(-1)^{m}\zeta\right),\quad
1\leq m\leq N,$ (4.33)

satisfies the ordinary differential equation of order $N$,

$\Bigg{[}\prod_{j=1}^{N}\bigg{(}\zeta\frac{d}{d\zeta}-\frac{\lambda_{j}}{N}\bigg{)}-\zeta\Bigg{]}\eta(\zeta)=0,\quad\zeta\in{\mathbb{C}}\backslash\\{0\\}.$
(4.34)

Given these preparations, Theorem 3.1 yields the following result:

######  Theorem 4.3.

Assume the notation established in (4.29)–(4.34). Then a fundamental system of
solutions of the $N$th-order differential equation (4.34) is given as follows:

$\displaystyle\eta_{1}(\zeta)$
$\displaystyle=G_{0,N}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{N};\frac{\lambda_{2}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}-\zeta\right)$
(4.37)
$\displaystyle=\frac{\zeta^{\lambda_{1}/N}}{\prod_{j^{\prime}=1}^{N}\Gamma\Big{(}1+\frac{\lambda_{1}-\lambda_{j^{\prime}}}{N}\Big{)}}\,{}_{0}F_{N-1}\Bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+\frac{\lambda_{1}-\lambda_{2}}{N},\dots,1+\frac{\lambda_{1}-\lambda_{N}}{N}}\end{array}\Bigg{|}\,\zeta\Bigg{)},$
(4.40) $\displaystyle\eta_{2}(\zeta)$
$\displaystyle=G_{0,N}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{N},\frac{\lambda_{2}}{N};\frac{\lambda_{3}}{N}\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}\,\zeta\right),$
(4.43) $\displaystyle\vdots$ $\displaystyle\eta_{n_{1}}(\zeta)$
$\displaystyle=G_{0,N}^{n_{1},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}}}{N};\frac{\lambda_{n_{1}+1}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}\,(-1)^{n_{1}}\zeta\right),$
(4.46) $\displaystyle\eta_{n_{1}+1}(\zeta)$
$\displaystyle=G_{0,N}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+1}}{N};\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}}}{N},\frac{\lambda_{n_{1}+2}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}-\zeta\right)$
(4.49)
$\displaystyle=\frac{\zeta^{\lambda_{n_{1}+1}/N}}{\prod_{j^{\prime}=1}^{N}\Gamma\Big{(}1+\frac{\lambda_{n_{1}+1}-\lambda_{j^{\prime}}}{N}\Big{)}}$
$\displaystyle\quad\times{}_{0}F_{N-1}\Bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+\frac{\lambda_{n_{1}+1}-\lambda_{2}}{N},\dots,1+\frac{\lambda_{n_{1}+1}-\lambda_{N}}{N}}\end{array}\Bigg{|}\,\zeta\Bigg{)},$
(4.52) $\displaystyle\eta_{n_{1}+2}(\zeta)$
$\displaystyle=G_{0,N}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+1}}{N},\frac{\lambda_{n_{1}+2}}{N};\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}}}{N},\frac{\lambda_{n_{1}+3}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}\,\zeta\right),$
(4.55) $\displaystyle\vdots$ (4.56) $\displaystyle\eta_{n_{1}+n_{2}}(\zeta)$
$\displaystyle=G_{0,N}^{n_{2},0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+1}}{N},\frac{\lambda_{n_{1}+2}}{N},\dots,\frac{\lambda_{n_{1}+n_{2}}}{N};\frac{\lambda_{n_{1}+n_{2}+1}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}\,(-1)^{n_{2}}\zeta\right),$
(4.59) $\displaystyle\vdots$
$\displaystyle\eta_{n_{1}+n_{2}+\cdots+n_{r-1}+1}(\zeta)$
$\displaystyle=G_{0,N}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}}{N};\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}+\cdots+n_{r-1}}}{N},\frac{\lambda_{n_{1}+\cdots+n_{r-1}+2}}{N},\dots,\frac{\lambda_{N}}{N}}\end{array}\Bigg{|}\,-\zeta\right)$
(4.62)
$\displaystyle=\frac{\zeta^{\lambda_{n_{1}+\cdots+n_{r-1}+1}/N}}{\prod_{j^{\prime}=1}^{N}\Gamma\Big{(}1+\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}-\lambda_{j^{\prime}}}{N}\Big{)}}$
$\displaystyle\times{}_{0}F_{N-1}\Bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle
1+\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}-\lambda_{2}}{N},\dots,1+\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}-\lambda_{N}}{N}}\end{array}\Bigg{|}\,\zeta\Bigg{)},$
(4.65) $\displaystyle\eta_{n_{1}+n_{2}+\cdots+n_{r-1}+2}(\zeta)$
$\displaystyle=G_{0,N}^{2,0}\Bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}}{N},\frac{\lambda_{n_{1}+\cdots+n_{r-1}+2}}{N};\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}+\cdots+n_{r-1}}}{N},}\end{array}$
(4.68) $\displaystyle\hskip
136.5733pt\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+\cdots+n_{r-1}+3}}{N},\dots,\frac{b_{N}}{N}}\end{array}\Bigg{|}\,\zeta\Bigg{)},$
(4.71) $\displaystyle\vdots$
$\displaystyle\eta_{n_{1}+n_{2}+\cdots+n_{r-1}+n_{r}}(\zeta)\equiv
y_{N}(\zeta)$
$\displaystyle=G_{0,N}^{n_{r},0}\Bigg{(}\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{n_{1}+\cdots+n_{r-1}+1}}{N},\dots,\frac{\lambda_{n_{1}+\cdots+n_{r-1}+n_{r}}}{N}\equiv\frac{\lambda_{N}}{N};}\end{array}$
(4.74) $\displaystyle\hskip
105.2751pt\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{N},\dots,\frac{\lambda_{n_{1}+\cdots+n_{r-1}}}{N}}\end{array}\Bigg{|}\,(-1)^{n_{r}}\zeta\Bigg{)};\quad\zeta\in{\mathbb{C}}\backslash\\{0\\}.$
(4.77)

In addition, the replacement $\zeta\longrightarrow\mu(z/N)^{N}$ in (4.56)
yields a fundamental system of solutions $y_{j}(\mu,\,\cdot\,)$, $1\leq j\leq
N$, of the generalized Euler eigenvalue problem (4.2).

######  Example 4.4.

Let $N=2$, $\lambda_{j}\in{\mathbb{C}}$, $j=1,2$, then

$\tau_{2}(\lambda_{1},\lambda_{2})=z^{-2}\bigg{[}z\frac{d}{dz}-\lambda_{1}\bigg{]}\bigg{[}z\frac{d}{dz}-\lambda_{2}\bigg{]},\quad
z\in{\mathbb{C}}\backslash\\{0\\},$ (4.78)

and the solutions of

$\tau_{2}(\lambda_{1},\lambda_{2})y(\mu,z)=\mu
y(\mu,z),\quad\mu\in{\mathbb{C}},\;z\in{\mathbb{C}}\backslash\\{0\\},$ (4.79)

are given as follows:
Case $(i)$: If
$[(\lambda_{1}-\lambda_{2})/2]\in{\mathbb{C}}\backslash{\mathbb{Z}}$, then

$\displaystyle\begin{split}y_{1}(\mu,z)&=G_{0,2}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{2};\frac{\lambda_{2}}{2}}\end{array}\Bigg{|}-\mu
z^{2}/4\right)\\\
&=\frac{\big{(}\mu^{1/2}z/2)^{\lambda_{1}}}{\Gamma\Big{(}1+\frac{\lambda_{1}-\lambda_{2}}{2}\Big{)}}{}_{0}F_{1}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle 1+\frac{\lambda_{1}-\lambda_{2}}{2}}\end{array}\Bigg{|}\,\mu
z^{2}/4\right)\\\
&=\big{(}\mu^{1/2}z/2\big{)}^{(\lambda_{1}+\lambda_{2})/2}I_{(\lambda_{1}-\lambda)/2}\big{(}\mu^{1/2}z\big{)},\\\
y_{2}(\mu,z)&=G_{0,2}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{2}}{2};\frac{\lambda_{1}}{2}}\end{array}\Bigg{|}-\mu
z^{2}/4\right)\\\
&=\frac{\big{(}\mu^{1/2}z/2)^{\lambda_{1}}}{\Gamma\Big{(}1+\frac{\lambda_{2}-\lambda_{1}}{2}\Big{)}}{}_{0}F_{1}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle 1+\frac{\lambda_{2}-\lambda_{1}}{2}}\end{array}\Bigg{|}\,\mu
z^{2}/4\right)\\\
&=\big{(}\mu^{1/2}z/2\big{)}^{(\lambda_{1}+\lambda_{2})/2}I_{(\lambda_{2}-\lambda_{1})/2}\big{(}\mu^{1/2}z\big{)};\quad
z\in{\mathbb{C}}\backslash\\{0\\}.\end{split}$ (4.80)

Case $(ii)$: If $[(\lambda_{1}-\lambda_{2})/2]\in{\mathbb{Z}}$, then

$\displaystyle\begin{split}y_{1}(\mu,z)&=G_{0,2}^{1,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{1}}{2};\frac{\lambda_{2}}{2}}\end{array}\Bigg{|}-\mu
z^{2}/4\right)\\\
&=\frac{\big{(}\mu^{1/2}z/2)^{\lambda_{1}}}{\Gamma\Big{(}1+\frac{\lambda_{1}-\lambda_{2}}{2}\Big{)}}{}_{0}F_{1}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle 1+\frac{\lambda_{1}-\lambda_{2}}{2}}\end{array}\Bigg{|}\,\mu
z^{2}/4\right)\\\
&=\big{(}\mu^{1/2}z/2\big{)}^{(\lambda_{1}+\lambda_{2})/2}I_{(\lambda_{1}-\lambda)/2}\big{(}\mu^{1/2}z\big{)},\\\
y_{2}(\mu,z)&=G_{0,2}^{2,0}\left(\\!\begin{array}[]{c}{\displaystyle}\hfil\\\
{\displaystyle\frac{\lambda_{2}}{2},\frac{\lambda_{1}}{2}}\end{array}\Bigg{|}-\mu
z^{2}/4\right)\\\
&=2\big{(}\mu^{1/2}z/2\big{)}^{(\lambda_{1}+\lambda_{2})/2}K_{(\lambda_{1}-\lambda_{2})/2}\big{(}\mu^{1/2}z\big{)};\quad
z\in{\mathbb{C}}\backslash\\{0\\}.\end{split}$ (4.81)

Here we used [13, No. 9.2.2] in case $(i)$ and [13, No. 8.4.8] in case $(ii)$.

##  5\. Some Remarks on the Nonhomogeneous Euler Differential Equation

In our final section we add some facts in connection with the inhomogeneous
$N$th-order Euler differential equations, that is, we consider the general
solution of the nonhomogeneous $N$th-order Euler differential equation.

######  Proposition 5.1.

Let $N\in{\mathbb{N}}$, $\lambda_{j}\in{\mathbb{C}}$, $1\leq j\leq N$, and
$f\in C({\mathbb{C}})$. Then the general solution of the nonhomogeneous $N$th-
order Euler differential equation

$\tau_{N}(\lambda_{1},\dots,\lambda_{N})y(z)=z^{-N}f(z),\quad\lambda_{j}\in{\mathbb{C}},\;z\in{\mathbb{C}}\backslash\\{0\\},$
(5.1)

is of the following form222For simplicity of notation in (5.3) we incorporated
the overall factor $z^{-N}$ on the left-hand side of (5.1) also into its
nonhomogeneous term on the right-hand side.: Grouping
$\\{\lambda_{1},\dots,\lambda_{N}\\}$ into pairwise distinct elements counting
multiplicity,

$\displaystyle\begin{split}&\\{\lambda_{1},\dots,\lambda_{N}\\}=\bigcup_{k=1}^{K}\\{\underbrace{\lambda_{j_{k}},\dots,\lambda_{j_{k}}}_{m_{j_{k}}\,times}\\},\quad\lambda_{j_{k}}\neq\lambda_{j_{k^{\prime}}},\;1\leq
k,k^{\prime}\leq K,\;k\neq k^{\prime},\;1\leq K\leq N,\\\
&\sum_{k=1}^{K}m_{j_{k}}=N,\end{split}$ (5.2)

the general solution of (5.1) is given by

$\displaystyle y(z)$
$\displaystyle=\sum_{k=1}^{K}\sum_{m=0}^{m_{j_{k}}-1}\big{\\{}C_{k,m}z^{\lambda_{j_{k}}}[\operatorname{ln}(z)]^{m}\big{\\}}+\int_{0}^{1}ds_{1}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}z\prod_{j=1}^{N}s_{j}\Bigg{)}\prod_{\ell=1}^{N}s_{\ell}^{-\lambda_{\ell}-1},$
$\displaystyle\hskip 85.35826ptC_{k,m}\in{\mathbb{C}},\;0\leq m\leq
m_{j_{k}}-1,\,1\leq k\leq K,\;z\in{\mathbb{C}}\backslash\\{0\\}.$ (5.3)

###### Proof.

That

$\displaystyle\begin{split}&\sum_{k=1}^{K}\sum_{m=0}^{m_{j_{k}}-1}\big{\\{}C_{k,m}z^{\lambda_{j_{k}}}[\operatorname{ln}(z)]^{m}\big{\\}},\\\
&\,C_{k,m}\in{\mathbb{C}},\;0\leq m\leq m_{j_{k}}-1,\,1\leq k\leq
K,\;z\in{\mathbb{C}}\backslash\\{0\\},\end{split}$ (5.4)

is the general solution of the homogeneous $N$th-order Euler differential
equation is well-known; it easily follows from taking repeated
$\lambda$-derivatives of the underlying indicial (or characteristic)
polynomial

$P_{N}(\lambda)=\prod_{j=1}^{N}(\lambda-\lambda_{j}),$ (5.5)

noting that ($z\in{\mathbb{C}}\backslash\\{0\\}$)

$\displaystyle\tau_{N}(\lambda_{1},\dots,\lambda_{N})\big{(}z^{\lambda}\big{)}=P_{N}(\lambda)\big{(}z^{\lambda+N}\big{)},$
$\displaystyle\tau_{N}(\lambda_{1},\dots,\lambda_{N})\big{(}(\partial/\partial\lambda)z^{\lambda}\big{)}=\tau_{N}(\lambda_{1},\dots,\lambda_{N})\big{(}z^{\lambda}\operatorname{ln}(z)\big{)}$
$\displaystyle\quad=(\partial/\partial\lambda)\big{[}\tau_{N}(\lambda_{1},\dots,\lambda_{N})\big{(}z^{\lambda}\big{)}\big{]}=[P_{N}^{\prime}(\lambda)+P_{N}(\lambda)\operatorname{ln}(z)]z^{\lambda+N},$
(5.6) $\displaystyle\quad\;\;\vdots$ etc.

Hence, it remains to show that

$\int_{0}^{1}ds_{1}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}z\prod_{j=1}^{N}s_{j}\Bigg{)}\prod_{\ell=1}^{N}s_{\ell}^{-\lambda_{\ell}-1},\quad
z\in{\mathbb{C}}\backslash\\{0\\},$ (5.7)

is a particular solution of (5.1). This in turn follows from iterating the
elementary computation,

$\displaystyle\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{[}\int_{0}^{1}ds_{1}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}z\prod_{j=1}^{N}s_{j}\Bigg{)}\prod_{\ell=1}^{N}s_{\ell}^{-\lambda_{\ell}-1}\Bigg{]}$
$\displaystyle\quad=\prod_{j=1}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{[}z^{\lambda_{1}}\int_{0}^{z}d\zeta_{1}\int_{0}^{1}ds_{2}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}\zeta_{1}\prod_{j=2}^{N}s_{j}\Bigg{)}$
$\displaystyle\hskip
241.84842pt\times\zeta_{1}^{-\lambda_{1}-1}\prod_{\ell=2}^{N}s_{\ell}^{-\lambda_{\ell}-1}\Bigg{]}$
$\displaystyle\quad=\prod_{j=2}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{[}(\lambda_{1}-\lambda_{1})z^{\lambda_{1}}\int_{0}^{z}d\zeta_{1}\int_{0}^{1}ds_{2}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}\zeta_{1}\prod_{j=2}^{N}s_{j}\Bigg{)}$
$\displaystyle\hskip
284.52756pt\times\zeta_{1}^{-\lambda_{1}-1}\prod_{\ell=2}^{N}s_{\ell}^{-\lambda_{\ell}-1}$
$\displaystyle\hskip
88.2037pt+z^{1+\lambda_{1}}\int_{0}^{1}ds_{2}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}z\prod_{j=2}^{N}s_{j}\Bigg{)}z^{-\lambda_{1}-1}\prod_{\ell=2}^{N}s_{\ell}^{-\lambda_{\ell}-1}\Bigg{]}$
$\displaystyle\quad=\prod_{j=2}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{[}\int_{0}^{1}ds_{2}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}z\prod_{j=2}^{N}s_{j}\Bigg{)}\prod_{\ell=2}^{N}s_{\ell}^{-\lambda_{\ell}-1}\Bigg{]}$
$\displaystyle\quad=\prod_{j=2}^{N}\bigg{[}z\frac{d}{dz}-\lambda_{j}\bigg{]}\Bigg{[}z^{\lambda_{2}}\int_{0}^{z}d\zeta_{2}\int_{0}^{1}ds_{3}\cdots\int_{0}^{1}ds_{N}\,f\Bigg{(}\zeta_{2}\prod_{j=3}^{N}s_{j}\Bigg{)}$
$\displaystyle\hskip
216.2411pt\times\zeta_{2}^{-\lambda_{2}-1}\prod_{\ell=3}^{N}s_{\ell}^{-\lambda_{\ell}-1}\Bigg{]}$
$\displaystyle\quad\;\;\vdots$
$\displaystyle\quad=\bigg{[}z\frac{d}{dz}-\lambda_{N}\bigg{]}z^{\lambda_{N}}\int_{0}^{z}d\zeta_{N}\,f(\zeta_{N})\zeta_{N}^{-\lambda_{N}-1}$
$\displaystyle\quad=(\lambda_{N}-\lambda_{N})z^{\lambda_{N}}\int_{0}^{z}d\zeta_{N}\,f(\zeta_{N})\zeta_{N}^{-\lambda_{N}-1}+z^{1+\lambda_{N}}f(z)z^{-\lambda_{N}-1}$
$\displaystyle\quad=f(z),\quad z\in{\mathbb{C}}\backslash\\{0\\}.$ (5.8)

Here the path from $0$ to $z$ in the $\zeta_{j}$-plane, $1\leq j\leq N$, can
be chosen to be a straight line segment. ∎

We note that the special case $N=2$ in Proposition 5.1 is mentioned in [10, p.
202].

Acknowledgments. We are indebted to Mark Ashbaugh and Andrei Martinez-
Finkelshtein for very helpful hints to the literature in connection with
Meijer’s $G$-function.

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