# Influence of geometrical perturbations on the ion motional frequencies in a
Paul trap

Manoj Kumar Joshi  Atomic and Molecular Physics Division, Bhabha Atomic
Research Centre, Mumbai 400085, India  Pushpa M. Rao<EMAIL_ADDRESS>Atomic and Molecular Physics Division, Bhabha Atomic Research Centre, Mumbai
400085, India

###### Abstract

The effect of geometrical perturbations on ion oscillation frequencies in a
$3$-D quadrupole trap is analyzed by theoretical simulations and compared with
the experimental results. Theoretically, the distorted potential within this
non-ideal trap caused by the insertion of filament has been simulated. An
analytical expression for the potential is then fitted and numerical solution
of the equations of trapped ion motion is obtained. Motional frequencies are
then calculated from the Fourier transformation of the simulated ion
trajectories at any given trapping potential and compared with our
experimental findings. The shift in the secular frequency with respect to the
level of insertion of the filament within the trap is evaluated.

Keywords: Quadrupole ion trap, Motional frequencies, Multipolar expansion,
Trapped ion trajectories

##  I Introduction

Confinement of charged particles in an ion trap offers unique advantageous
allowing state of the art experiments to be performed, resulting in a profound
insight in the fields of atomic, nuclear and quantum physics. Ion traps are
extensively used in the different fields of scientific applications like mass
spectrometry, several atomic and nuclear physics related experiments and more
recently in the field of quantum computations Savard and Werth (2000); Blaum
(2006); Thompson (1993); Monroe et al. (1996). Starting with Paul Paul (1990)
and Dehmelt Dehmelt (1968) several other researchers Brown and Gabrielse
(1986); Prestage et al. (1991); Werth (1996) have extensively worked in this
field presenting innovative and improved models of ion traps. Improvement in
design of ion traps and theoretical simulations to achieve the near ideal
quadrupole field is an ongoing evolutionary process in the field.
Conceptually, trapping of charged particles is not a difficult task but
utilization of traps as devices for a quantum engineering, precise mass
measurements and high precision studies in atomic and molecular physics,
requires an appropriate design and thorough understanding of the ion traps. In
the past thirty years, an enormous work has been done in the field of ion
traps.

For an ideal $3$-D Paul trap the field inside varies linearly with distance
from the trap centre. And potential inside is given as

$\phi=\frac{U_{dc}+V_{rf}cos(\varOmega t)}{2r_{0}^{2}}(2z^{2}-x^{2}-y^{2}),$
(1)

where $U_{dc}$ and $V_{rf}$ are the amplitudes of the static and dynamic
trapping potentials applied across the trap electrodes and $\varOmega$ is the
angular frequency of the RF field. The size of trap is defined in terms of
radius of ring electrode ($r_{0}$) and the separation of end-caps ($2z_{0}$)
and the relation $r_{0}/z_{0}$ equal to $\surd 2$ is maintained. The ion
motion for an ideal trap can be described by Mathieu equations as discussed in
Major et al. (2005); March (2005). In practice, it is difficult to make an
ideal trap that is described in theory. Manufacturing errors and several other
modifications made in the trap for the convenience of specific experiments
cause deviations in the potential given in equation (1). Any deviation from
this ‘ideal’ quadrupole potential is expressed either by multipole expansion
of the generalised potential J. D. Jackson (1975) or suitable modifications in
the equation (1) Wang and Franzen (1992a). This deviation of the potential
from the ideal quadrupole form alters the characteristics of the trapped ions,
manifesting as nonlinear effects which are broadly classified in three ways.

The first one is where the trapped ions exhibit instabilities even though the
trap is operated within the stability region. These instabilities were first
seen by Busch and Paul Busch and Paul (1961) and were termed as black holes
and canyons. Measurement of the number of trapped ions at different operating
points provides an experimental verification Alheit et al. (1995, 1996a);
Vedel et al. (1998) of the occurrence of these instabilities. Wang et al Wang
et al. (1993) have given a theoretical explanation for such instabilities
wherein they say that at certain operating points the superposition of the
motional frequencies and their overtones are an integral multiple of the drive
frequency, causing the transfer of energy from the drive field to the trapped
ions.

The second effect usually seen is the nonlinear resonance occurring in the
resonant excitation and ejection of trapped ions due to coupling of ion
motion. This causes the excitation and ejection of trapped ions at frequencies
other than the harmonic motional frequencies Wang and Franzen (1992b); Chu et
al. (1998); Vedel and Vedel (1990). These nonlinear effects are used in
isotope separation of ions in a Paul trap Alheit et al. (1996b).

The third observed effect in ion traps is the shift in the secular frequencies
due to nonlinearities in the trapping field. As ions move away from the centre
of a non-ideal trap, where the field within varies nonlinearly with distance,
the secular frequencies exhibit a shift from those calculated for an ideal
trap. The shift in the ion oscillation frequency depends upon the polarity,
strength and order of that particular multipole present Franzen (1991). This
geometrically driven nonlinear effect exhibits a shift in the stability region
and is observed in hybrid Arkin et al. (1999) and stretched mode Johnson et
al. (1992) of Paul traps. This shift in the stability region is explainable
either by numerical methods Sevugarajan and Menon (2001); Zhou et al. (2010)
or by including weak nonlinear potentials in the Mathieu equations that
describe the motion of trapped ions Sevugarajan and Menon (2002).

For an ideal trap the motional frequencies depend solely upon the mass,
operating parameters and trap dimensions. The motional frequencies get
modified due to the presence of other ions and aberrations in the trap
geometry. Several researchers have shown that when ions are collectively
excited the motional resonances do not depend on the number of ions trapped
ions i.e., no observable shift in the motional resonance frequencies is seen
due to the space charge effect Alheit et al. (1996b); Paasche et al. (2002).
The perturbations in the trapping field due to geometrical aberrations are the
main cause of the shift in motional frequencies. These nonlinearities affect
high precision studies which rely on the accurate measurements of motional
frequencies.

In almost all cases the potential within a non-ideal trap is considered to be
rotationally symmetric. In the current work we discuss the change in the
trapped ion characteristics when they are subjected to a slightly asymmetric
potential caused due to geometric modifications. In our experiment the
filament which is placed slightly above the lower end cap, alters the
potential within the trap and also causes the breakage of rotational symmetry.
Cumulative effects of the breakage of rotational symmetry and alteration of
the trapping potential lead to changes in the trapped ion motion. In this
article we mainly discuss the shift in motional frequencies due to the
geometry driven perturbations through our experimental and theoretical
studies.

##  II Theory of trapping

In an ideal Paul trap, the time dependent potential given in equation (1) is
responsible for confining the ions close to the trap centre. The motion of
trapped ion is described by Mathieu equations McLachlan (1947) and is
expressed as

$\frac{d^{2}u_{i}}{d\tau^{2}}+(a_{i}-2q_{i}cos2\tau)u_{i}=0,$ (2)

where $u_{i}$ represents the position coordinate in Cartesian coordinate
system and the normalized time $\tau$ is $\varOmega t/2$. The parameters,
$a_{i}$ and $q_{i}$ are the functions of trapping potentials and are defined
as

$a_{z}=\frac{8QU_{dc}}{mr_{0}^{2}\varOmega^{2}},a_{x}=a_{y}=\frac{-4QU_{dc}}{mr_{0}^{2}\varOmega^{2}},$
(3)
$q_{z}=\frac{-4QV_{rf}}{mr_{0}^{2}\varOmega^{2}},q_{x}=q_{y}=\frac{2QV_{rf}}{mr_{0}^{2}\varOmega^{2}},$
(4)

where $m$ and $Q$ are the mass and charge of trapped species respectively.

The solutions of equation (2) represent the motion of ion in the trap
comprised of the secular motion modulated at the drive frequency
($\varOmega$). The secular motional frequency ($\omega_{i}$) depends on the
mass, charge of trapped species and operating parameters of the trap. In the
pseudo potential well model Dehmelt (1968), the secular frequency $\omega_{i}$
is expressed as function of a parameter $\beta_{i}$, which is a function of
$a_{i}$ and $q_{i}$ defined in equation (3) and (4). For small values of
$a_{i}$ and $q_{i}$ ($<0.4$) the parameter $\beta_{i}$ is approximated to
$\sqrt{a_{i}+q_{i}^{2}/2}$. In other cases $\beta_{i}$ can be written as a
solution of a continuous function in $a_{i}$ and $q_{i}$ Major et al. (2005);
March (2005). These expressions are applicable only for an ideal Paul trap and
any imperfection calls for modification in the expressions.

Figure 1:  Orthogonal view of the trap geometry with filament raised.

##  III  Experiment

The experimental set up used in current studies is similar to that reported in
our earlier paper Bhattacharyya et al. (2006). The ion trap consists of a set
of three hyperbolically shaped electrodes. The ring electrode radius ($r_{0}$)
is $20$mm. The ring electrode has four symmetrically positioned holes each
measuring $8$mm in diameter to facilitate steering of the laser beam Joshi and
Rao (2012). The lower end-cap has two rectangular slots to place filaments
within the trap( figure 1). In our experiments a filament of size
$22\text{mm}\times 8\text{mm}$ is placed $5$mm above and $6$mm away from the
centre of lower end-cap. For trapping, a RF potential at fixed drive frequency
($\varOmega/2\pi=500$kHz) along with the DC potential was applied to the ring
electrode keeping end-caps at the ground potential. The DC and RF potentials
were chosen in a way such that the operating parameters ($a_{i}$, $q_{i}$) are
well within the stability region. Europium ($Eu^{+}$) ions were loaded in the
trap by surface ionization of the sample placed on the filament. The base
vacuum attained in the trap assembly was $5\times 10^{-9}$torr and the ions
were continuously loaded in the trap during the experiment.

The trapped ions are non-destructively detected Bhattacharyya et al. (2006) by
excitation of the motional frequencies using a weak RF field applied across a
high impedance tank circuit ($2$M$\varOmega$) at its resonance frequency. This
tank circuit is connected across the end-caps of the trap. Figure (2) shows
the schematic of the trapping and detection setup. The ion oscillation
frequency is swept across the excitation frequency by varying the DC potential
($U_{dc}$). At resonance, damping of the excitation voltage across the tank
circuit takes place and this signal is recorded. The number of ions confined
in the trap is estimated from the fitting of the signal obtained to the
response signal of an equivalent circuitry of trapped ions Gaboriaud et al.
(1981). In our experiment the number of ions confined is estimated to be $\sim
10^{6}$.

Figure 2:  Schematic diagram of the ion trapping setup and detection circuit.

##  IV Theory and simulation

Using SIMION software Manual (2009), the potential generated within the ion
trap used in our experimental studies was theoretically simulated. The 3D
potential generated is then fitted to an analytical expression which is a
series expansion of the generalized potential considered up to the quadrupole
term and is given as

$\phi=\phi_{0}\left(\alpha_{0}+\frac{\alpha_{1}x+\gamma_{1}z}{r_{0}}+\frac{\alpha_{2}x^{2}+\beta_{2}y^{2}+\gamma_{2}z^{2}}{r_{0}^{2}}\right),$
(5)

where $\phi_{0}$ is the potential applied at the ring electrode with the end-
caps electrically grounded. Cartesian coordinates $x$, $y$ and $z$ are the
position coordinates within the trapping volume considering the trap center as
origin. In the above equation the coefficients $\alpha$, $\beta$ and $\gamma$
represent the strength of the multipolar terms of the potential. In accordance
with our experiments, the theoretical simulations have been carried out with
the filament kept at the same potential as the end-caps. From figure (1) it is
clear that our trap has ‘$XZ$’ plane of symmetry, thus the coefficient for the
dipolar term or any higher order odd terms along the $y$-coordinate can be
neglected throughout the simulations. Trapped ion trajectories are computed by
solving the equations of motion numerically using Runge Kutta $4$ code with
the potential given in equation (5). Motional frequencies are then evaluated
from the Fourier transformation of the simulated ion trajectories for a given
set of trapping potentials. We repeat the simulations for different heights of
the filament.

##  V Result and discussion

The filament inside the ion trap acts as an extra electrode and causes a
deviation in the potential which affects the motional behavior of the trapped
ions. To study these effects experimentally, we follow the set of operating
potentials ($V_{rf}$ and $U_{dc}$) for which we see the resonant excitation
signal at $55.6$kHz. In figure (3) traces $\\#1$ and $\\#2$ correspond to the
experimental set of operating potential that follow the axial and radial equi-
frequency lines respectively. For comparison, we evaluated the set of
potentials for an ideal trap using pseudo potential model, for which the axial
and radial motional frequencies are $55.6$kHz. In figure (3) traces $\\#3$ and
$\\#4$ are the evaluated set of potentials ($V_{rf}$, $U_{dc}$) in the axial
and radial modes respectively. From figure (3) it is clear that the sets of
operating potentials, at which the damping of resonant excitation signal was
observed, differ drastically from the sets of potentials evaluated based on
the Pseudo potential model. This clearly shows that the experimental ion
oscillation frequencies differ from those calculated for an ideal trap. We
have seen that the calculated ion oscillation frequency for a given set of
potential at which resonant excitation signal was seen ($\\#1$ and $\\#2$ in
figure 3), differs by approximately $14$kHz from the excitation frequency.
This calls for a modification of the expression used to theoretically
calculate the ion oscillation frequencies.

Figure 3:  Plot shows theoretical and experimental equi-frequency lines in
radial and axial directions. In this plot $\\#1$ and $\\#2$ are the
experimentally observed equi-frequency set of potentials in axial and radial
direction respectively. And $\\#3$ and $\\#4$ are the set of potentials
calculated based upon Pseudo potential model for an ideal trap.

For further analysis, we simulated the potential generated within the ion trap
using SIMION and fit it to an analytical expression given in equation (5). The
coefficients of the least square fit are shown in table 1. Substituting these
coefficients in equation (5), the single ion trajectories are simulated for
the same set of operating potentials at which we have observed the trapped ion
signal. The ion oscillation frequencies both in the radial and axial direction
are evaluated from the Fourier transformation of the simulated trajectories.

Table 1: Coefficients calculated from the least square fit. Coefficie- | Calculated | Error
---|---|---
nts | values |  estimated ($\pm$)
$\alpha_{0}$ | -0.1037 | 0.0008
$\alpha_{1}$ | -0.0823 | 0.0012
$\gamma_{1}$ | 0.2047 | 0.0014
$\alpha_{2}$ | 0.6902 | 0.0041
$\beta_{2}$ | 0.5500 | 0.0041
$\gamma_{2}$ | -1.2405 | 0.0064

Simulations show a deviation of the potential generated within the trap used
in our experiments from an ideal quadrupole potential. It can be seen that the
potential within the trap electrodes does not exhibit rotational symmetry.
This breakage of rotational symmetry is evident from Table 1, as the
coefficients $\alpha_{2}$ and $\beta_{2}$ are not equal, $\alpha_{1}$ and
$\gamma_{1}$ coefficients are not zero. The simulated ion oscillation
frequencies in $x$ and in $y$ directions are thus seen to be non equal
($\omega_{x}\neq\omega_{y}$). For instance, the simulated motional frequencies
for a given set of operating potentials ($V_{rf}=700$V, $U_{dc}=-5.3$V) are
$\omega_{x}=53.5$kHz; $\omega_{y}=41.73$kHz and $\omega_{r}=53.5$kHz.

Frequencies in radial ($\omega_{r}$) and axial ($\omega_{z}$) directions were
simulated at several sets of experimental operating potentials and are
compared with the fixed excitation frequency used in the current experiment.
In figure (4) the line $\\#$c represents the excitation frequency ($55.6$kHz)
which is compared with the simulated axial and radial ion oscillation
frequencies, shown in lines $\\#$a and $\\#$b, for the set of operating
potentials used in the experiment. In figure (4) each number on the $x-$axis
represents a set of operating potentials ($V_{rf}$, $U_{dc}$) in increasing
order of $V_{rf}$ values, corresponding to the equi-frequency lines $\\#1$ and
$\\#2$ of figure (3). The $x-$axis in figure (4) maps the set of operating
potentials shown along the line $\\#1$ and $\\#2$ in figure (3) and the
simulated frequencies are plotted along the $y-$axis. From figure (4) it is
seen that the evaluated axial and radial motional frequencies are slightly
different from the experimental excitation frequency and also exhibit an
opposite shift. This can be attributed to the non zero contribution of the
higher order multipoles (higher than the quadrupole term) which is not taken
into account in the present theoretical simulations.

Figure 4:  Plot between simulated motional frequencies and operating
potentials at which trapped ion signal was observed along the equi-frequency
line. In this plot $\\#$a and $\\#$b are set of potentials that follows the
equi-frequency lines in axial and radial direction respectively versus the
simulated ion oscillation frequency. And line $\\#$c represents the excitation
frequency at $55.6$kHz.

It is seen that the observed shift in the motional frequencies increases with
increasing $q_{z}$ ($V_{rf}$ ) values and this is attributed to the increase
in the kinetic energy of the trapped ions Siemers et al. (1988). With
increasing kinetic energy the trapped ions move away from the trap centre,
leading to a more pronounced effect owing to the contributions of the higher
order multipoles. Franzen et al and Sevugarajan et al in their articles
Franzen (1991); Sevugarajan and Menon (2002) have theoretically described
similar effects on the radial and axial motions due to the higher order
multipole contributions (specifically due to the octopole term).

To see the extent of geometrical aberrations and its effect on the ion
oscillation frequencies, the axial ion oscillation frequencies were evaluated
at different filament heights keeping the operating potential fixed at
$V_{rf}=550$V, $U_{dc}=21.75$V. Figure (5) shows a plot of the ion oscillation
frequencies simulated at filament heights varied from ${0-6}$mm. From this
figure it is clear that the ion oscillation frequency increases as a function
of filament height. This is because as the filament is moved towards the trap
centre, the changes in the potential are more pronounced.

Figure 5:  Calculated secular ion oscillation frequency in the axial direction
plotted at different filament heights.

##  VI Conclusion

In the present study we have seen the effect of geometrical aberrations in
Paul trap on the ion oscillation frequencies. Due to the insertion of
filament, the ion trap used is no longer ‘ideal’ and the potential within is
also not a pure quadrupole potential. Thus the ion oscillation frequencies
evaluated for an ideal trap is no longer valid. The change in the potential
shifts the motional frequencies and this has been studied both experimentally
and theoretically.

Using dipolar excitation, we have been able to resonantly excite and detect
the ion oscillations, both in axial and radial directions. This is possible
because of the presence of a small quadrupolar contribution in dipolar mode of
excitation. Experimental studies wherein equi-frequency lines are traced at
several set of operating potentials within the stability region differ
drastically from the equi-frequency lines calculated for an ideal quadrupole
trap.

Theoretically, we have simulated the near exact form of the potential within
the trap with geometrical aberrations caused by insertion of the filament.
From simulations we see that the coefficients calculated from the multipolar
expansion for any given order differs from that of an ideal trap. In addition,
a breakage of rotational symmetry is also observed. The motional frequencies
evaluated from the simulations for this non-ideal trap is seen to be slightly
shifted from the experimental resonant excitation frequency. Further, the ion
oscillation frequencies have been simulated for different filament heights. We
see that the simulated axial ion oscillation frequency increases as the
filament is moved towards the trap centre.

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# Analytic continuations of the Horn $H_{1}$ and $H_{5}$ functions

Souvik Bera Tanay Pathak souvikbera<EMAIL_ADDRESS>
(Centre for High Energy Physics, Indian Institute of Science,
Bangalore-560012, Karnataka, India )

###### Abstract

The analytic continuations (ACs) of the double variable Horn $H_{1}$ and
$H_{5}$ functions have been derived for the first time using the Mathematica
package Olsson.wl. The corresponding region of convergences (ROCs) of the ACs
are obtained using a companion package ROC2.wl. A Mathematica package
HornH1H5.wl, containing all the derived ACs and the associated ROCs, along
with a demonstration file of the same is publicly available in this URL :
https://github.com/souvik5151/Horn_H1_H5.git

##  1 Introduction

The theory of hypergeometric functions and their ACs in one and more than one
variable is now highly explored subject for over a century and is of
fundamental importance in many branches of mathematical physics. [1, 2, 3, 4,
5, 6]. The one variable Gauss ${}_{2}F_{1}$ function have been generalized to
two variables in [7], which are now known as Appell $F_{1},F_{2},F_{3}$ and
$F_{4}$ functions. Another ten double variable hypergeometric functions
$G_{1},G_{2},G_{3}$ and $H_{1},H_{2},\dots,H_{7}$ are introduced by Horn [8].
These fourteen Appell-Horn functions form the set of distinct, second order
complete hypergeometric functions in two variables. More general higher-order
functions, namely the Kampé de Fériet functions, frequently appear in the
study of analytic continuations of these functions [2]. Using the detailed
transformation theory of two variables hypergeometric functions it was found
by Erdélyi [9] that all the second order, two variable hypergeometric
functions, except $F_{4},H_{1}$ and $H_{5}$ can be related to $F_{2}$ via the
transformation formulae. The Appell $F_{2}$ has been studied by Olsson[10] and
recently have also been studied in [11] where a large number of ACs and their
numerical implementation in Mathematica has been developed. The Appell $F_{4}$
was treated by Exton [12], and these results were further modified in [13]. It
was also pointed out by Exton that $H_{1}$ and $H_{5}$ should be investigated
in a similar way [12]. A recent study for the determination of the solution of
the PDEs of various Appell and Horn function along with that of $H_{1}$ and
$H_{5}$ have been done in [14]. Some properties of the $H_{1}$ and $H_{5}$
have recently been studied in [15, 16, 17, 18, 19]. This serves as a strong
motivation for the present work where we derive the ACs and their
corresponding ROC for $H_{1}$ and $H_{5}$.

Erdélyi pointed out that the well-known transformation formulae of the one
variable ${}_{p}F_{p-1}$ functions can be used to find the ACs of the
multivariate hypergeometric functions [9]. The method has been used for the
Appell $F_{1}$ [20], $F_{4}$ [12] and recently for $F_{2}$[11]. We see that
the similar investigation for the case of Horn $H_{1}$ and $H_{5}$ is
considerably difficult as compared of other case previously analyzed. It is
also worthwhile to mention that the analysis of ROC in these cases is
complicated, in comparison with $F_{4}$ and $F_{2}$, specifically for the case
of $H_{5}$. These occur due to the higher order hypergeometric functions that
appear in these studies. We therefore use the Mathematica package Olsson.wl
[21] for the derivation of the ACs of $H_{1}$ and $H_{5}$. This package is
helpful in deriving the linear transformations of the multivariable
hypergeometric function using the various transformation properties of single
variable hypergeometric functions listed in Appendix 6.1. It also includes
another companion package ROC2.wl, which automatizes the Horn’s theorem for
finding the ROC of two-variable hypergeometric functions. A brief overview of
Horn’s theorem to find the region of convergence can be found in Appendix 6.2.
These packages eases the calculation of derivation of the various ACs and
their corresponding ROC which can be difficult or at times not possible to
compute by hand, in an efficient manner.

The aim of this work is to provide the ACs for $H_{1}$ and $H_{5}$ that cover
the whole real $x-y$ plane and their respective ROCs, exceptional parameters
being excluded. While deriving the ACs, we observe that only the use of AC of
${}_{p}F_{p-1}(z)$ around $z=\infty$ and the AC of ${}_{2}F_{1}(z)$ around
$z=1$ are not adequate. To resolve this issue we use the Pfaff-Euler
transformations (PETs) of the ${}_{2}F_{1}$, wherever applicable, which allow
us to cover the regions that are otherwise difficult to cover. For the case of
$H_{1}$ we found that such approach allows us to cover the whole real plane,
apart from some boundaries and singular points. On the other hand for the the
Horn $H_{5}$ function there is still remains a small finite size region where
no AC is valid, (see Figure 29).

Since it is impossible to present all the results in the form of running text,
we provide along with this manuscript a Mathematica [22] package and a
demonstration file of the package. The package can be used to access the ACs
and their corresponding ROCs as required by the user. However we point out
that special care has to be taken for the numerical evaluation as the ACs can
have multivaluedness issues which are not discussed in the present version.
The scheme of this paper is as follows: In section 2 we have first analyzed
$H_{1}$, starting with the derivation of one of its simple AC for the
illustration. All the other ACs are derived using the package Olsson.wl that
follows the same methodology. Similarly, we have discussed $H_{5}$ in section
3. In all these sections, we have discussed ways along with a road map to
derive enough ACs that cover the whole real plane (boundaries excluded). We
have also illustrated the importance and uses of the Pfaff-Euler
transformation of ${}_{2}F_{1}$ hypergeometric function during the process.

##  2 The Horn $H_{1}$ function

The Horn $H_{1}$ function is defined as [8, 2]

$\displaystyle
H_{1}(a,b,c,d,x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{m-n}(b)_{m+n}(c)_{n}x^{m}y^{n}}{m!n!(d)_{m}}$
(1)

with the ROC :
$2\sqrt{\left|x\right|\left|y\right|}+\left|y\right|<1\land\left|y\right|<1\land\left|x\right|<1$,
which for real x and y is shown in Figure.2

It satisfies the following system of partial differential equations

$\displaystyle x(1-x)r-y^{2}t+[d-(a+b+1)x]p-(a-b-1)yq-abz=0$ (2)
$\displaystyle-y(1-y)t-x(1-y)s+[a-1-(b+c+1)y]q-cxp-cbz=0$

where the symbols are defined as follows

$\displaystyle z=H_{1}\hskip 14.22636pt,\hskip 14.22636ptp=\frac{\partial
z}{\partial x}\hskip 14.22636pt,\hskip 14.22636ptq=\frac{\partial z}{\partial
y}$ $\displaystyle r=\frac{\partial^{2}z}{\partial x^{2}}\hskip
14.22636pt,\hskip 14.22636ptt=\frac{\partial^{2}z}{\partial y^{2}}\hskip
14.22636pt,\hskip 14.22636pts=\frac{\partial^{2}z}{\partial x\partial y}$

The singular curves of the above equation are as follows, see Figure.2

$x=0,y=0,x=1,y=-1,-4xy+y^{2}+2y+1=0$ (3)

Figure 1: ROC of Eq. (1)

Figure 2: Singular curve of $H_{1}$

In this compilation we will find the various AC of the $H_{1}$ function by
using the ACs of $\,{}_{p}F_{p-1}$. We will also make use of the Pfaff-Euler
transformations of ${}_{2}F_{1}$ in the intermediate steps and show how to
obtain the various other ACs that could not be or are at times are hard to
obtain using just the ACs of ${}_{p}F_{p-1}$.

###  2.1 An Illustrative example

As an illustrative example of the procedure of finding the AC, by deriving one
simple AC of $H_{1}$. We will derive the AC around $\mathbf{(1,0)}$.

We take Eq. (1) and sum over m,

$\displaystyle
H_{1}(a,b,c,d,x,y)=\sum_{m,n=0}^{\infty}\frac{(-1)^{n}y^{n}(b)_{n}(c)_{n}}{n!(1-a)_{n}}\,_{2}F_{1}(b+n,a-n;d;x)$
(4)

Now using the AC of ${}_{2}F_{1}(x)$ around $x=1$ Eq. (58) we find two series,

$\displaystyle H_{(1,0)}^{(1)}(a,b,c,d,x,y)$
$\displaystyle=\dfrac{\Gamma(d)\Gamma(-a-b+d)}{\Gamma(d-a)\Gamma(d-b)}\sum_{m,n=0}^{\infty}\frac{y^{n}(b)_{n}(c)_{n}(b-d+1)_{n}}{n!(1-a)_{n}(d-a)_{n}}$
$\displaystyle\times\,_{2}F_{1}(a-n,b+n;a+b-d+1;1-x)$
$\displaystyle=\dfrac{\Gamma(d)\Gamma(-a-b+d)}{\Gamma(d-a)\Gamma(d-b)}\sum_{m,n}\frac{(c)_{n}(a)_{m-n}(b-d+1)_{n}(b)_{m+n}}{m!n!(d-a)_{n}(a+b-d+1)_{m}}(1-x)^{m}(-y)^{n}$
(5)

with ROC :
$2\sqrt{\left|1-x\right|\left|-y\right|}+\left|-y\right|<1\land\left|1-x\right|<1\land\left|-y\right|<1$

and,

$\displaystyle H_{(1,0)}^{(2)}(a,b,c,d,x,y)=$
$\displaystyle(1-x)^{-a-b+d}\dfrac{\Gamma(d)\Gamma(a+b-d)}{\Gamma(a)\Gamma(b)}\sum_{m,n=0}^{\infty}\frac{y^{n}(c)_{n}}{n!}$
$\displaystyle\times\,_{2}F_{1}(-b+d-n,-a+d+n;-a-b+d+1;1-x)$
$\displaystyle=(1-x)^{-a-b+d}\dfrac{\Gamma(d)\Gamma(a+b-d)}{\Gamma(a)\Gamma(b)}$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{(c)_{n}(b-d+1)_{n}(d-a)_{m+n}(d-b)_{m-n}}{m!n!(d-a)_{n}(-a-b+d+1)_{m}}(1-x)^{m}(-y)^{n}$
(6)

So we get following AC around (1,0)

$H_{1}(a,b,c,d,x,y)=\dfrac{\Gamma(d)\Gamma(-a-b+d)}{\Gamma(d-a)\Gamma(d-b)}\sum_{m,n=0}^{\infty}\frac{(c)_{n}(a)_{m-n}(b-d+1)_{n}(b)_{m+n}}{m!n!(d-a)_{n}(a+b-d+1)_{m}}(1-x)^{m}(-y)^{n}\\\
\\\
+(1-x)^{-a-b+d}\dfrac{\Gamma(d)\Gamma(a+b-d)}{\Gamma(a)\Gamma(b)}\sum_{m,n=0}^{\infty}\frac{(c)_{n}(b-d+1)_{n}(d-a)_{m+n}(d-b)_{m-n}}{m!n!(d-a)_{n}(-a-b+d+1)_{m}}(1-x)^{m}(-y)^{n}$
(7)

with ROC :
$2\sqrt{\left|1-x\right|\left|-y\right|}+\left|-y\right|<1\land\left|1-x\right|<1\land\left|-y\right|<1$
as shown in Figure.3

Figure 3: ROC of Eq. (7)

We observe that around the singular point $(1,0)$ one AC is enough to cover
the space around it. However there might arise situation where this is not
possible. For example for the other singular point $(0,-1)$, where we need two
ACs around that singular point.
The following diagram illustrates the possible chain of process that can be
used to derive both the ACs around the singular point $(0,-1)$

$(0,0)$$(0,-1)$First AC$(0,-1)$Second AC

Using the AC of ${}_{p}F_{p-1}(z)$ around $z=\infty$, we have the following
two possible situations

$(0,0)$$(0,\infty)$$(\infty,\infty)$

$(0,0)$$(\infty,0)$$(\infty,\infty)$

Taking into account all these possibilities we can have in general following
possibilities for $H_{1}$.

$(0,0)$$\textbf{T}_{0}$$(\infty,0)$$\textbf{T}_{4}$$(\infty,\infty)$$\textbf{T}_{9}$$(1,0)$$\textbf{T}_{1}$$(\infty,0)$$\textbf{T}_{5}$$(\infty,\infty)$$\textbf{T}_{11}$$(1,\infty)$$\textbf{T}_{8}$$(\infty,\infty)$$\textbf{T}_{13}$$(0,\infty)$$\textbf{T}_{6}$$(\infty,\infty)$$\textbf{T}_{10}$$(0,-1)$$\textbf{T}_{2}$$(0,-1)$$\textbf{T}_{3}$$(0,\infty)$$\textbf{T}_{7}$$(\infty,\infty)$$\textbf{T}_{12}$$(\infty,-1)$$\textbf{T}_{s1}$$(\infty,\infty)$$\textbf{T}_{s2}$
Figure 4: All the possibilities of ACs for $H_{1}$. The thick lines indicates
the one that have been derived and dashed lines denotes which are not. The
region that could be covered using the ACs denoted by dashed lines are already
covered using the other ACs. The derived ACs are part of package and the AC
$T_{i}$ in the above graph is the $i-th$ AC in the package.

###  2.2 The Pfaff-Euler transformations

In the intermediate steps of the analytic continuation procedure, it is
sometimes possible to get ${}_{2}F_{1}$ hypergeometric function when carrying
out one of the summation. Instead of using the usual AC of ${}_{2}F_{1}(z)$
around $z=1$ or $z=\infty$ it is also possible to use the PETs, Eq. (57), in
the intermediate steps. This process can also be done in Olsson.wl. Using this
we get the following transformations for $H_{1}$

We first take Eq. (1). Summing over m, we find,

$\displaystyle
H_{1}(a,b,c,d,x,y)=\sum_{m,n=0}^{\infty}\frac{(-1)^{n}y^{n}(b)_{n}(c)_{n}}{n!(1-a)_{n}}\,_{2}F_{1}(b+n,a-n;d;x)$
(8)

Now using the PETs of ${}_{2}F_{1}$, we find,

$\displaystyle
H_{1}(a,b,c,d,x,y)=(1-x)^{-b}\sum_{m,n=0}^{\infty}\frac{(c)_{n}(b)_{m+n}(d-a)_{m+n}}{m!n!(1-a)_{n}(d)_{m}(d-a)_{n}}\left(\frac{x}{x-1}\right)^{m}\left(\frac{y}{x-1}\right)^{n}$
(9)

with ROC :
$\sqrt{\left|\frac{y}{x-1}\right|}+\sqrt{\left|\frac{x}{x-1}\right|}<1$

Figure 5: ROC of Eq. (9)

Figure 6: ROC of Eq. (10)

$\displaystyle
H_{1}(a,b,c,d,x,y)=(1-x)^{-a}\sum_{m,n=0}^{\infty}\frac{(b)_{n}(c)_{n}(a)_{m-n}(b-d+1)_{n}(d-b)_{m-n}}{m!n!(d)_{m}}\left(\frac{x}{x-1}\right)^{m}(y(x-1))^{n}$
(10)

with ROC :
$\sqrt{\left|\frac{x}{x-1}\right|}-\frac{1}{\sqrt{\left|y(x-1)\right|}}<-1\land\left|y(x-1)\right|<1\land\left|\frac{x}{x-1}\right|<1$

$\displaystyle
H_{1}(a,b,c,d,x,y)=(1-x)^{-a-b+d}\sum_{m,n=0}^{\infty}\frac{(b)_{n}(c)_{n}(b-d+1)_{n}(d-a)_{m+n}(d-b)_{m-n}}{m!n!(1-a)_{n}(d)_{m}(d-a)_{n}}x^{m}y^{n}$
(11)

with the ROC same as $H_{1}$ :
$2\sqrt{\left|x\right|\left|y\right|}+\left|y\right|<1\land\left|y\right|<1\land\left|x\right|<1$

Figure 7: ROC of Eq. (11)

Figure 8: ROC of Eq. (13)

Similarly, summing over n, we find,

$\displaystyle
H_{1}(a,b,c,d,x,y)=\sum_{m,n=0}^{\infty}\frac{x^{m}(a)_{m}(b)_{m}}{m!(d)_{m}}\,_{2}F_{1}(c,b+m;-a-m+1;-y)$
(12)

Using PETs of ${}_{2}F_{1}$ we find,

$\displaystyle
H_{1}(a,b,c,d,x,y)=(y+1)^{-b}\sum_{m,n=0}^{\infty}\frac{(a+c)_{m}(b)_{m+n}(-a-c+1)_{n-m}}{m!n!(d)_{m}(1-a)_{n-m}}\left(\frac{x}{y+1}\right)^{m}\left(\frac{y}{y+1}\right)^{n}$
(13)

with ROC $\left|\frac{x}{y+1}\right|+\left|\frac{y}{y+1}\right|<1$

$\displaystyle
H_{1}(a,b,c,d,x,y)=(y+1)^{-c}\sum_{m,n=0}^{\infty}\frac{(b)_{m}(c)_{n}(a+b)_{2m}(-a-b+1)_{n-2m}}{m!n!(d)_{m}(1-a)_{n-m}}(-x)^{m}\left(\frac{y}{y+1}\right)^{n}$
(14)

with ROC :
$\frac{1}{\left|-x\right|}-4\left|\frac{y}{y+1}\right|^{2}>4\left|\frac{y}{y+1}\right|\land\left|x\right|<1\land\left|\frac{y}{y+1}\right|<1$

$\displaystyle H_{1}(a,b,c,d,x,y)$ $\displaystyle=(y+1)^{-a-b-c+1}$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{(b)_{m}(a+b)_{2m}(a+c)_{m}(-a-b+1)_{n-2m}(-a-c+1)_{n-m}}{m!n!(d)_{m}(1-a)_{n-m}}\left(\dfrac{x}{(y+1)^{2}}\right)^{m}(-y)^{n}$
(15)

with ROC
$\frac{1}{\sqrt{\left|\frac{x}{(y+1)^{2}}\right|}}-\left|-y\right|>1\land\left|\frac{x}{(y+1)^{2}}\right|<1\land\left|-y\right|<1$

Figure 9: The ROC of Eq. (14)

Figure 10: The ROC of Eq. (2.2)

It is important to note that though the PETs has been used with the original
definition of $H_{1}$ in the above case. One can also use them in the
intermediate steps, if summation over one of the indices give ${}_{2}F_{1}$.
One AC of $H_{1}$ that is labeled by $22$ has been derived using this
strategy.
One can also then use the whole procedure of finding the AC on the above
derived transformations of $H_{1}$ and can get many more ACs of $H_{1}$. We
derive AC number $20$ and $21$ using this idea.

##  3 The Horn $H_{5}$ function

The Horn function $H_{5}$ is defined as [8, 1, 2],

$\displaystyle
H_{5}:=H_{5}(a,b,c,x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{2m+n}(b)_{n-m}}{(c)_{n}}\frac{x^{m}y^{n}}{m!n!}$
(16)

The defining ROC is given by [2]

$\displaystyle|x|<\frac{1}{4}\land|y|<1\land$
$\displaystyle|y|<\left(\begin{array}[]{cc}\\{&\begin{array}[]{cc}\min\left(\frac{-(1-12|x|)^{3/2}+36|x|+1}{54|x|},\frac{(1-12|x|)^{3/2}+36|x|+1}{54|x|},\frac{(12|x|+1)^{3/2}-36|x|+1}{54|x|}\right)&\sqrt{|x|}<\frac{1}{2\sqrt{3}}\\\
\frac{(12|x|+1)^{3/2}-36|x|+1}{54|x|}&\text{True}\\\ \end{array}\\\
\end{array}\right)$ (20)

The ROC above is shown in Figure. 12 for real values of $x$ and $y$.

Figure 11: The defining ROC of $H_{5}$ (Eq. (3)) is plotted for real values of
$x$ and $y$.

Figure 12: The singular curves (i.e. Eq. (22) are shown

The Horn $H_{5}$ function satisfy the following set of PDEs

$\displaystyle
x(1+4x)r+(4x-1)ys+y^{2}t+\left((4a+6)x-b+1\right)p+2(a+1)yq+a(1+a)z=0$
$\displaystyle(y-1)yt+xys-2x^{2}r-(x(a-2b+2))p+(y(a+b+1)-c)q+abz=0$ (21)

where as before the symbols $p,q,r,t$ and $s$ are as follows

$\displaystyle z=H_{5}\hskip 14.22636pt,\hskip 14.22636ptp=\frac{\partial
z}{\partial x}\hskip 14.22636pt,\hskip 14.22636ptq=\frac{\partial z}{\partial
y}$ $\displaystyle r=\frac{\partial^{2}z}{\partial x^{2}}\hskip
14.22636pt,\hskip 14.22636ptt=\frac{\partial^{2}z}{\partial y^{2}}\hskip
14.22636pt,\hskip 14.22636pts=\frac{\partial^{2}z}{\partial x\partial y}$

It is to be noted that the PDE of $H_{5}$ given in the book [1] is not
correct. This observation is also noted in [14]. Some properties of the Horn
$H_{5}$ function are studied in [16].

The singular curves for the Horn $H_{5}$ are as follows

$\displaystyle x=0,y=0,x=-1/4,y=1,1+8x+16x^{2}-y-36xy+27xy^{2}=0$ (22)

These are shown in Figure 12.

In the following section, we find the ACs of the function using the Olsson’s
method. All the ACs are derived using the package Olsson.wl [21]. The ROCs of
the ACs are obtained using the companion package ROC2.wl. The ROCs that are
shown below are plotted for real values of $x$ and $y$.

To proceed, we take the summation over the index $m$ and observe that the
Gauss ${}_{2}F_{1}$ appears inside the summand,

$\displaystyle
H_{5}(a,b,c,x,y)=\sum_{n=0}^{\infty}\frac{y^{n}(a)_{n}(b)_{n}}{n!(c)_{n}}\,_{2}F_{1}\left(\frac{a}{2}+\frac{n}{2},\frac{a}{2}+\frac{n}{2}+\frac{1}{2};-b-n+1;-4x\right)$
(23)

Similarly taking the summation over the index $n$, we find

$\displaystyle
H_{5}(a,b,c,x,y)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{m}(a)_{2m}}{m!(1-b)_{m}}\,_{2}F_{1}(b-m,a+2m;c;y)$
(24)

The above two expressions (i.e. Eq. (23) and Eq. (24)) are the starting points
for our analysis of the $H_{5}$ function. Applying the well known linear
transformation formulae of Gauss ${}_{2}F_{1}$ listed in the Section 6.1, the
ACs of $H_{5}$ can be found for general values of its Pochhammer parameters.

Let us take Eq. (24). Applying Eq. (58), we find the AC of $H_{5}$ around the
point $(0,1)$,

$\displaystyle H_{5}=$
$\displaystyle\frac{\Gamma(c)\Gamma(-a-b+c)}{\Gamma(c-a)\Gamma(c-b)}\sum_{m,n=0}^{\infty}\frac{(a-c+1)_{2m}(a)_{2m+n}(b)_{n-m}}{(c-b)_{m}(a+b-c+1)_{m+n}}\frac{(-x)^{m}(1-y)^{n}}{m!n!}$
$\displaystyle+(1-y)^{-a-b+c}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\sum_{m,n=0}^{\infty}\frac{(a-c+1)_{2m}(c-a)_{n-2m}(c-b)_{m+n}}{m!n!(c-b)_{m}(-a-b+c+1)_{n-m}}\left(-\frac{x}{1-y}\right)^{m}(1-y)^{n}$
(25)

The first series of the above expression (Eq. (3)) is convergent in

$\displaystyle|x|<\frac{1}{16}\land|1-y|<1\land|1-y|<\frac{(1-12|x|)^{3/2}-18|x|+1}{54|x|}$

whereas the second series is valid for

$\displaystyle\left|\frac{x}{y-1}\right|<1\land|1-y|<1\land$
$\displaystyle|1-y|<\left(\begin{array}[]{cc}\\{&\begin{array}[]{cc}\min\left(\frac{1}{27}\left(-\frac{(4|x|-3|y-1|)^{3/2}}{\left|\sqrt{x}(y-1)\right|}-\frac{8|x|}{|y-1|}+9\right),\frac{1}{27}\left(\frac{(4|x|+3|y-1|)^{3/2}}{\left|\sqrt{x}(y-1)\right|}-\frac{8|x|}{|y-1|}-9\right)\right)&4\left|\frac{x}{y-1}\right|>3\\\
\frac{1}{27}\left(\frac{(4|x|+3|y-1|)^{3/2}}{\left|\sqrt{x}(y-1)\right|}-\frac{8|x|}{|y-1|}-9\right)&\text{True}\\\
\end{array}\\\ \end{array}\right)$ (29)

Hence the AC (Eq. (3)) is valid in the common ROCs of the two series. We plot
the ROCs of the individual series in Figure 14 and 14.

Figure 13: The ROC of the first series of Eq. (3)

Figure 14: The ROC of the second series of Eq. (3)

We observe that the AC of $H_{5}$ in Eq. (3) does not cover the whole space
around the singular point $(0,1)$. Thus we can find another AC that will cover
the space that remains uncovered. To find such AC we take the second series of
Eq. (3) (let us denote it by $S_{2}$) and take the summation over $m$
explicitly

$\displaystyle S_{2}=$
$\displaystyle(1-y)^{-a-b+c}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty}\frac{(1-y)^{n}(c-a)_{n}(c-b)_{n}}{n!(-a-b+c+1)_{n}}$
$\displaystyle\times\,_{4}F_{3}\left(\frac{a}{2}-\frac{c}{2}+\frac{1}{2},\frac{a}{2}-\frac{c}{2}+1,a+b-c-n,-b+c+n;c-b,\frac{a}{2}-\frac{c}{2}-\frac{n}{2}+\frac{1}{2},\frac{a}{2}-\frac{c}{2}-\frac{n}{2}+1;\frac{x}{1-y}\right)$
(30)

Now using the AC of ${}_{4}F_{3}(\dots;z)$ at $z=\infty$ (Eq. (61)), we find
that only one series is non-vanishing. Denoting it as $S_{2}^{\prime}$

$\displaystyle S_{2}^{\prime}$
$\displaystyle=(1-y)^{-a-b+c}\left(\frac{x}{y-1}\right)^{-a-b+c}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}$
$\displaystyle\times\sum_{m,n=0}^{\infty}\frac{(a+2b-2c+1)_{m-n}(a+b-c)_{m-n}(a+2b-c)_{2m-n}}{m!n!(a+2b-2c+1)_{m-2n}(a+2b-c)_{2(m-n)}}\left(\frac{1-y}{x}\right)^{m}(-x)^{n}$
(31)

whose ROC is given by

$\displaystyle|r|<1\land\sqrt{|s|}<\text{min}\left(z_{1},z_{2},z_{3}\right)\land|s|<\frac{1}{16}$
(32)

where $r=\frac{1-y}{x}$, $s=-x$ and

$\displaystyle z_{1}=$ $\displaystyle\text{~{}second root of
~{}}27z^{4}|r|^{2}-2z^{2}(9|r|+8)-|r|-1=0$ $\displaystyle z_{2}=$
$\displaystyle\text{~{}second root of
~{}}27z^{4}|r|^{2}+2z^{2}(9|r|-8)+|r|-1=0$ $\displaystyle z_{3}=$
$\displaystyle\text{~{}second root of
~{}}27z^{4}|r|^{2}+2z^{2}(9|r|+8)-|r|-1=0$

This ROC is plotted in Figure 16.

Hence we find the second AC of $H_{5}$ around the point $(0,1)$ by combining
the first series of Eq. (3) with $S_{2}^{\prime}$ (i.e. Eq. (3))

$\displaystyle H_{5}=$
$\displaystyle\frac{\Gamma(c)\Gamma(-a-b+c)}{\Gamma(c-a)\Gamma(c-b)}\sum_{m,n=0}^{\infty}\frac{(a-c+1)_{2m}(a)_{2m+n}(b)_{n-m}}{(c-b)_{m}(a+b-c+1)_{m+n}}\frac{(-x)^{m}(1-y)^{n}}{m!n!}$
$\displaystyle+(1-y)^{-a-b+c}\left(\frac{x}{y-1}\right)^{-a-b+c}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}$
$\displaystyle\times\sum_{m,n=0}^{\infty}\frac{(a+2b-2c+1)_{m-n}(a+b-c)_{m-n}(a+2b-c)_{2m-n}}{m!n!(a+2b-2c+1)_{m-2n}(a+2b-c)_{2(m-n)}}\left(\frac{1-y}{x}\right)^{m}(-x)^{n}$
(33)

The ROC of this AC is same as Eq. (32), which is plotted in Figure 16.

Figure 15: The ROC of series $S_{2}^{\prime}$ (i.e. Eq. (32)) is plotted for
real values of $x$ and $y$.

$(0,0)$$(0,1)$Eq. (3)$(0,1)$Eq. (3) Figure 16: The procedure to find the ACs
around $(0,1)$ is described as a directed graph

Let us summarise what we have achieved so far. We started with the definition
of the $H_{5}$ whose ROC is given in Eq. (3) and plotted in Figure 12. Taking
summation over the index $n$ yields the Gauss ${}_{2}F_{1}$ in the summand.
Then good use of the AC of ${}_{2}F_{1}$ Eq. (58) is used to find the AC of
$H_{5}$ at $(0,1)$. The ROC of that AC plotted in Figure 14 for real values of
$x,y$. Since the AC of $H_{5}$ around $(0,1)$ does not cover the whole space
around that point, we have derived another AC around the same point by
transforming the second series of Eq. (3). The resultant AC is given in Eq.
(3) and the ROC of it is plotted in Figure 16. This procedure can be described
by a directed graph as shown in Figure 16, where the vertices represent the
singular points and the directed edges denotes the process of evaluation. We
plot the ROCs of both the ACs around the point $(0,1)$ along with the defining
ROC of $H_{5}$ in Figure 18.

Figure 17: The defining ROC of $H_{5}$, the ROC of first AC around $(0,1)$
(i.e. Eq. (3)) and the ROC of second AC around $(0,1)$ (Eq. (3)) are plotted
in blue, yellow and green respectively for real values of $x,y$.

$(0,0)$$(0,\infty)$Eq. (3)$(\infty,\infty)$Eq. (3) Figure 18: The procedure to
find the ACs around $(0,\inf)$ and $(\infty,\infty)$ are described as a
directed graph.

We go on to derive the ACs of $H_{5}$ around $(0,\infty)$ and
$(\infty,\infty)$. The procedure is demonstrated via a directed graph in
Figure 18. To find the AC around $(0,\infty)$, we use Eq. (59) in Eq. (24) to
find

$\displaystyle H_{5}$
$\displaystyle=(-y)^{-a}\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\sum_{m,n=0}^{\infty}\frac{(a)_{2m+n}(a-c+1)_{2m+n}}{m!n!(a-b+1)_{3m+n}}\left(-\frac{x}{y^{2}}\right)^{m}\left(\frac{1}{y}\right)^{n}$
$\displaystyle+(-y)^{-b}\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}\sum_{m,n=0}^{\infty}\frac{(b)_{n-m}(b-c+1)_{n-m}}{m!n!(-a+b+1)_{n-3m}}\left(-xy\right)^{m}\left(\frac{1}{y}\right)^{n}$
(34)

The ROC of the first series is given by

$\displaystyle
16\left|\frac{x}{y^{2}}\right|<27\land\left|\frac{x}{y^{2}}\right|<\frac{1}{32}\left(\left(9-\frac{8}{|y|}\right)^{3/2}-\frac{36}{|y|}+\frac{8}{|y|^{2}}+27\right)\land\frac{1}{|y|}<1$
(35)

and the second series is valid in

$\displaystyle 27|xy|<1\land|y|<y^{2}\land$
$\displaystyle|xy|<\min\left(\frac{1}{32}\left(\left(9-\frac{8}{|y|}\right)^{3/2}-\frac{36}{|y|}+\frac{8}{y^{2}}+27\right)|y|^{3},\frac{1}{32}|y|\left(-\sqrt{|y|}(9|y|+8)^{3/2}+36|y|+27y^{2}+8\right)\right)$
(36)

These are plotted in Figure 20 and 20 respectively. The ROC of the AC given in
Eq. (3) is the intersection of the ROCs of both the series.

Figure 19: The ROC of the first series of Eq. (3)

Figure 20: The ROC of the second series of Eq. (3)

We observe that, the second series can be transformed further to find the AC
of $H_{5}$ around $(\infty,\infty)$ in a similar way to the second AC around
$(0,1)$. We find the AC around $(\infty,\infty)$ as

$\displaystyle H_{5}=$
$\displaystyle=(-y)^{-a}\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\sum_{m,n=0}^{\infty}\frac{(a)_{2m+n}(a-c+1)_{2m+n}}{m!n!(a-b+1)_{3m+n}}\left(-\frac{x}{y^{2}}\right)^{m}\left(\frac{1}{y}\right)^{n}$
$\displaystyle+(-y)^{-b}(-xy)^{\frac{b-a}{3}}\frac{\Gamma(1-b)\Gamma(c)\Gamma\left(\frac{a-b}{3}\right)}{3\Gamma(a)\Gamma\left(-\frac{a}{3}-\frac{2b}{3}+1\right)\Gamma\left(-\frac{a}{3}+c-\frac{2b}{3}\right)}\times$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{\left(\frac{a}{3}-\frac{b}{3}\right)_{m-\frac{n}{3}}\left(\frac{a}{3}+\frac{2b}{3}\right)_{m+\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+1\right)_{m+\frac{2n}{3}}\left(\frac{1}{27xy}\right)^{m}\left(-\frac{\sqrt[3]{-xy}}{y}\right)^{n}}{m!n!\left(\frac{1}{3}\right)_{m}\left(\frac{2}{3}\right)_{m}\left(-\frac{a}{3}-\frac{2b}{3}+1\right)_{-\frac{1}{3}(2n)}\left(\frac{a}{3}+\frac{2b}{3}\right)_{\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+1\right)_{\frac{2n}{3}}\left(-\frac{a}{3}+c-\frac{2b}{3}\right)_{-\frac{2n}{3}}}$
$\displaystyle-\frac{2\pi
3^{-a+b-\frac{1}{2}}(-y)^{-b}\Gamma(1-b)\Gamma(c)\Gamma(a-b)(-xy)^{\frac{1}{3}(-a+b-1)}}{\Gamma(a)\Gamma\left(\frac{1}{3}(-a-2b+2)\right)\Gamma\left(\frac{a-b}{3}\right)\Gamma\left(\frac{1}{3}(a-b+2)\right)\Gamma\left(\frac{1}{3}(-a-2b-1)+c\right)}\times$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{\left(\frac{a}{3}-\frac{b}{3}+\frac{1}{3}\right)_{m-\frac{n}{3}}\left(\frac{a}{3}+\frac{2b}{3}+\frac{1}{3}\right)_{m+\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+\frac{4}{3}\right)_{m+\frac{2n}{3}}\left(\frac{1}{27xy}\right)^{m}\left(-\frac{\sqrt[3]{-xy}}{y}\right)^{n}}{m!n!\left(\frac{2}{3}\right)_{m}\left(\frac{4}{3}\right)_{m}\left(-\frac{a}{3}-\frac{2b}{3}+\frac{2}{3}\right)_{-\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}+\frac{1}{3}\right)_{\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+\frac{4}{3}\right)_{\frac{2n}{3}}\left(-\frac{a}{3}+c-\frac{2b}{3}-\frac{1}{3}\right)_{-\frac{2n}{3}}}$
$\displaystyle+\frac{\pi
3^{-a+b-\frac{1}{2}}(-y)^{-b}\Gamma(1-b)\Gamma(c)\Gamma(a-b)(-xy)^{\frac{1}{3}(-a+b-2)}}{\Gamma(a)\Gamma\left(\frac{1}{3}(-a-2b+1)\right)\Gamma\left(\frac{a-b}{3}\right)\Gamma\left(\frac{1}{3}(a-b+1)\right)\Gamma\left(-\frac{a}{3}+c-\frac{2(b+1)}{3}\right)}$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{\left(\frac{a}{3}-\frac{b}{3}+\frac{2}{3}\right)_{m-\frac{n}{3}}\left(\frac{a}{3}+\frac{2b}{3}+\frac{2}{3}\right)_{m+\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+\frac{5}{3}\right)_{m+\frac{2n}{3}}\left(\frac{1}{27xy}\right)^{m}\left(-\frac{\sqrt[3]{-xy}}{y}\right)^{n}}{m!n!\left(\frac{4}{3}\right)_{m}\left(\frac{5}{3}\right)_{m}\left(-\frac{a}{3}-\frac{2b}{3}+\frac{1}{3}\right)_{-\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}+\frac{2}{3}\right)_{\frac{2n}{3}}\left(\frac{a}{3}+\frac{2b}{3}-c+\frac{5}{3}\right)_{\frac{2n}{3}}\left(-\frac{a}{3}+c-\frac{2b}{3}-\frac{2}{3}\right)_{-\frac{2n}{3}}}$
(37)

We do not write the ROC of the above AC explicitly. It can be obtained from
the ancillary file (See Section 6.3). The ROC is shown in Figure 22.

Figure 21: The ROC of the Eq. (3)

Figure 22: The ROCs of the four ACs that are obtained so far.

To summarize, we have found four ACs of $H_{5}$ around the singular points
$(0,1),(0,\infty)$ and $(\infty,\infty)$. Two ACs are found around the
neighborhood of $(0,1)$. The ROCs of these four ACs are plotted in Figure 22.
In an analogous way, starting from Eq. (23) one can find ACs around $(1,0)$,
which can further be used to find ACs around $(\infty,0),(1,\infty)$ and
$(\infty,\infty)$. The whole procedure of finding ACs of $H_{5}$ is
demonstrated using the directed graph in Figure 23. However finding all the
ACs according to the graph is quite a laborious task in practise, due to the
lengthy expressions of the ACs, even when one uses the computer program
Olsson.wl. On the other hand, one may use the PETs of the Gauss ${}_{2}F_{1}$
Eq. (57) to find some ACs of $H_{5}$. We now show examples of how to find such
ACs.

$(0,0)$$\mathbf{S_{1}}$$(\infty,0)$$\mathbf{S_{12}}$$(\infty,\infty)$$(1/4,0)$$\mathbf{S_{10}}$$(1/4,0)$$\mathbf{S_{11}}$$(\infty,0)$$\mathbf{S_{13}}$$(\infty,\infty)$$(1/4,\infty)$$(\infty,\infty)$$(0,\infty)$$\mathbf{S_{4}}$$(\infty,\infty)$$\mathbf{S_{6}}$$(0,1)$$\mathbf{S_{2}}$$(0,1)$$\mathbf{S_{3}}$$(0,\infty)$$\mathbf{S_{5}}$$(\infty,\infty)$$\mathbf{S_{7}}$$(\infty,1)$$\mathbf{S_{8}}$$(\infty,\infty)$$\mathbf{S_{9}}$
Figure 23: The graph shows all the possibilities of obtaining ACs for $H_{5}$.
The thick lines indicates the one that have been derived and dashed lines
denotes which are not. The ROCs of the ACs that can be derived following the
dashed lines overlap with the ROC of the obtained $13$ ACs. The other $7$ ACs
that are not mentioned in the above graph are derived using the PETs of
${}_{2}F_{1}$.

As before, out starting points are Eq. (23) and Eq. (24). Let us start with
Eq. (23). Using the first PET (i.e. first equation of Eq. (57)) we find,

$\displaystyle
H_{5}=(4x+1)^{-\frac{a}{2}}\sum_{m,n=0}^{\infty}\frac{(a)_{n}\left(\frac{a}{2}\right)_{m+\frac{n}{2}}\left(\frac{1}{2}(-a-2b+1)\right)_{m-\frac{3n}{2}}}{m!n!\left(\frac{a}{2}\right)_{\frac{n}{2}}(c)_{n}\left(\frac{1}{2}(-a-2b+1)\right)_{-\frac{1}{2}(3n)}(1-b)_{m-n}}\left(\frac{4x}{4x+1}\right)^{m}\left(-\frac{y}{\sqrt{4x+1}}\right)^{n}$
(38)

This AC is valid in

$\displaystyle\left\\{|r|<1\land|s|<\frac{1}{4}\land|s|<\left(\begin{array}[]{cc}\\{&\begin{array}[]{cc}\min\left(\Phi_{1}(r),\Phi_{2}(r),\Phi_{3}(r)\right)&|r|<\frac{1}{4}\\\
\Phi_{3}(r)&\text{True}\\\ \end{array}\\\ \end{array}\right)\right\\}$ (42)

where $r=\frac{4x}{4x+1},~{}s=-\frac{y^{2}}{16x+4}$. The functions
$\Phi_{i}(r)$ are defined below

$\displaystyle\Phi_{1}(r)$
$\displaystyle=\frac{-128|r|^{3}+96|r|^{2}+3|r|-2\sqrt{1-|r|}\sqrt{-(4|r|-1)^{3}}(8|r|+1)+2}{729|r|^{2}}$
$\displaystyle\Phi_{2}(r)$
$\displaystyle=\frac{-128|r|^{3}+96|r|^{2}+3|r|+2(8|r|+1)\sqrt{1-|r|}\sqrt{-(4|r|-1)^{3}}+2}{729|r|^{2}}$
$\displaystyle\Phi_{3}(r)$
$\displaystyle=\frac{-2\sqrt{|r|+1}|1-8|r||(4|r|+1)^{3/2}+128|r|^{3}+96|r|^{2}-3|r|+2}{729|r|^{2}}$
(43)

Similarly using other two PETs, we find two more ACs. These are written below

$\displaystyle
H_{5}=(4x+1)^{\frac{1}{2}(-a-1)}\sum_{m,n=0}^{\infty}\frac{(a)_{n}\left(\frac{a+1}{2}\right)_{m+\frac{n}{2}}\left(-\frac{a}{2}-b+1\right)_{m-\frac{3n}{2}}}{m!n!\left(\frac{a+1}{2}\right)_{\frac{n}{2}}(c)_{n}\left(-\frac{a}{2}-b+1\right)_{-\frac{1}{2}(3n)}(1-b)_{m-n}}\left(\frac{4x}{4x+1}\right)^{m}\left(-\frac{y}{\sqrt{4x+1}}\right)^{n}$
(44)

Figure 24: The ROC of the Eq. (38) and Eq. (44)

Figure 25: The ROCs of the Eq. (45)

We observe that the characteristic list of both the ACs Eq. (38) and Eq. (44)
are same. Hence the AC Eq. (44) is valid in the same ROC (i.e. Eq. (42)). The
other AC can be found using the last PET (i.e. the last equation of Eq. (57)

$\displaystyle
H_{5}=(4x+1)^{-a-b+\frac{1}{2}}\sum_{m,n=0}^{\infty}\frac{(a)_{n}\left(\frac{1}{2}(-a-2b+1)\right)_{m-\frac{3n}{2}}\left(-\frac{a}{2}-b+1\right)_{m-\frac{3n}{2}}\left(-4x\right)^{m}\left(-\frac{y}{(4x+1)^{2}}\right)^{n}}{m!n!(c)_{n}\left(\frac{1}{2}(-a-2b+1)\right)_{-\frac{1}{2}(3n)}\left(-\frac{a}{2}-b+1\right)_{-\frac{1}{2}(3n)}(1-b)_{m-n}}$
(45)

$\displaystyle|r|<1\land\sqrt{|s|}<\left(\begin{array}[]{cc}\\{&\begin{array}[]{cc}\min\left(\Psi_{1}(r),\Psi_{2}(r),\Psi_{3}(r),\Psi_{4}(r)\right)&\sqrt{|r|}<\frac{1}{\sqrt{3}}\\\
\min\left(\Psi_{3}(r),\Psi_{4}(r)\right)&\text{True}\\\ \end{array}\\\
\end{array}\right)\land|s|<\frac{1}{4}$ (49)

where $r=-4x,~{}s=\frac{y^{2}}{4(4x+1)^{4}}$. The functions $\Psi_{i}(r)$ are
defined below

$\displaystyle\Psi_{1}(r)$
$\displaystyle=\frac{-(1-3|r|)^{3/2}+9|r|+1}{27\left(|r|^{3/2}+\sqrt{|r|}\right)^{2}}\hskip
14.22636pt,\hskip
14.22636pt\Psi_{3}(r)=\frac{3|r|\left(\sqrt{3|r|+1}-3\right)+\sqrt{3|r|+1}+1}{27(|r|-1)^{2}|r|}$
$\displaystyle\Psi_{2}(r)$
$\displaystyle=\frac{(1-3|r|)^{3/2}+9|r|+1}{27\left(|r|^{3/2}+\sqrt{|r|}\right)^{2}}\hskip
14.22636pt,\hskip
14.22636pt\Psi_{4}(r)=\frac{\sqrt{3|r|+1}+3|r|\left(\sqrt{3|r|+1}+3\right)-1}{27(|r|-1)^{2}|r|}$
(50)

Starting with Eq. (24) and using three PETs we find three ACs below

$\displaystyle
H_{5}=(1-y)^{-b}\sum_{m,n=0}^{\infty}\frac{(a)_{2m}(a-c+1)_{2m}(b)_{n-m}(c-a)_{n-2m}}{m!n!(c)_{n}}\left(x-xy\right)^{m}\left(\frac{y}{y-1}\right)^{n}$
(51)

with ROC

$\displaystyle 4|x-xy|<1\land\left|\frac{y}{y-1}\right|<1\land$
$\displaystyle|x-xy|<\frac{\sqrt{\left|\frac{y}{y-1}\right|+8}\left|\frac{y}{y-1}\right|^{3/2}+\left|\frac{y}{y-1}\right|^{2}-20\left|\frac{y}{y-1}\right|+8\sqrt{\left|\frac{y}{y-1}\right|}\sqrt{\left|\frac{y}{y-1}\right|+8}-8}{32\left(\left|\frac{y}{y-1}\right|-1\right)^{3}}$
(52)

and

Figure 26: The ROC of the Eq. (51)

Figure 27: The ROCs of the Eq. (53)

$\displaystyle
H_{5}=(1-y)^{-a}\sum_{m,n=0}^{\infty}\frac{(a)_{2m+n}(c-b)_{m+n}}{m!n!(1-b)_{m}(c)_{n}(c-b)_{m}}\left(-\frac{x}{(y-1)^{2}}\right)^{m}\left(\frac{y}{y-1}\right)^{n}$
(53)

with ROC

$\displaystyle 4\left|\frac{x}{(y-1)^{2}}\right|<1\land
32\left|\frac{x}{(y-1)^{2}}\right|+\sqrt{\left|\frac{y}{y-1}\right|}\left(\left|\frac{y}{y-1}\right|+8\right)^{3/2}+\left|\frac{y}{y-1}\right|^{2}<20\left|\frac{y}{y-1}\right|+8\land\left|\frac{y}{y-1}\right|<1$
(54)

and the last one

$\displaystyle
H_{5}=(1-y)^{-a-b+c}\sum_{m,n=0}^{\infty}\frac{(a)_{2m}(a-c+1)_{2m}(c-a)_{n-2m}(c-b)_{m+n}}{m!n!(1-b)_{m}(c)_{n}(c-b)_{m}}\left(\frac{x}{y-1}\right)^{m}y^{n}$
(55)

with ROC

$\displaystyle
4\left|\frac{x}{y-1}\right|<1\land|y|<1\land\frac{\left(4\left|\frac{x}{y-1}\right|+3\right)^{3/2}}{\sqrt{\left|\frac{x}{y-1}\right|}}+8\left|\frac{x}{y-1}\right|>9(3|y|+2)$
(56)

All the ROCs of the six ACs are shown in Figure 25, 25, 27, 27 and 29 for real
values of $x$ and $y$.

It worth noting that, the PETs of ${}_{2}F_{1}$ can be used in any
intermediate step of the derivation, when the Gauss ${}_{2}F_{1}$ appears
inside the summand, to find new ACs. We have applied it on the definition of
the $H_{5}$ function only in this section to demonstrate the usefulness of the
PETs of the Gauss ${}_{2}F_{1}$.

Even after analysing all the possible ways of applying the PETs in the
intermediate steps of the derivation, we found no AC that is valid inside the
small ‘white’ region as shown in Figure 29.

Figure 28: The ROC of the Eq. (55)

Figure 29: The plot shows the uncovered region for real $x,y$.

##  4 Summary

In this work we have presented the various ACs and their corresponding ROCs of
the Horn $H_{1}$ and $H_{5}$ functions. These result have been derived using
the publicly available automated Mathematica packages Olsson.wl and ROC2.wl.
This ensures an easeful computation of the ACs and their ROCs which are
otherwise prone to errors or difficult to compute by hand.
Our extensive work shows that the ACs of $H_{1}$ cover the whole real $x-y$
plane, with the exception of few boundaries and singular lines, whereas the
ACs of $H_{5}$ can be used to cover everywhere, except for the small region
shown in Figure 29. It is to be noted that in the both cases we are not able
to give convergent expressions on the singular curves. It may be pointed out
that the expressions given here have to be used carefully for numerical
implementation due to problem of multi-valuedness, which has not been
discussed in this article. To this extent it is the precise analog of the work
of Exton for Appell $F_{4}$ and Olsson for Appell $F_{1}$. All the ACs derived
here are valid for generic values of Pochhammer parameters. If the ACs are
required to be used for the exceptional parameters then one has to do the
proper limiting procedure.

It is our view that investigations such as those reported here will go a long
way in the advancements of the theory of hypergeometric functions, even for
higher orders and higher number of variable, which in general are ubiquitous.
Furthermore, the use of various symbolic computational packages used here
makes them significant for the analysis of complicated hypergeometric
functions on a more firm footing.

##  5 Acknowledgements

The authors would like to thank B. Ananthanarayan for the discussions and
comments on the draft. We also thank Samuel Friot for his continual
encouragement and having initiated us into this subject.

##  6 Appendix

###  6.1 Appendix A

In this Appendix we list some formulae that are used in deriving the ACs of
$H_{1}$ and $H_{5}$ functions.

  * •

Euler transformation of ${}_{2}F_{1}$

$\begin{split}{}_{2}F_{1}(a,b,c;z)&=(1-z)^{-a}{}_{2}F_{1}\Big{(}a,c-b,c,\frac{z}{z-1}\Big{)}\\\
\\\ &=(1-z)^{-b}{}_{2}F_{1}\Big{(}c-a,b,c,\frac{z}{z-1}\Big{)}\\\ \\\
&=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b,c,z)\end{split}$ (57)

  * •

Analytic continuation of ${}_{2}F_{1}$ around $z=1$

$_{2}F_{1}(a,b,c;z)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\,_{2}F_{1}(a,b,a+b-c+1;1-z)\\\
+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}\,_{2}F_{1}(c-a,c-b,c-a-b+1;1-z)$
(58)

  * •

Analytic continuation of ${}_{2}F_{1}$ around $z=\infty$

$_{2}F_{1}(a,b,c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}(-z)^{-a}\,_{2}F_{1}(a,a-c+1,a-b+1;\frac{1}{z})\\\
+\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}(-z)^{-b}\,_{2}F_{1}(b,b-c+1,b-a+1;\frac{1}{z})$
(59)

  * •

Analytic continuation of ${}_{3}F_{2}$ around $z=\infty$

$_{3}F_{2}\left(\begin{matrix}a_{1},&a_{2},&a_{3}\\\ \\\
&b_{1},&b_{2}\end{matrix}\Bigg{|}{z}\right)=\\\
\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(a_{2}-a_{1})\Gamma(a_{3}-a_{1})}{\Gamma(a_{2})\Gamma(a_{3})\Gamma(b_{1}-a_{1})\Gamma(b_{2}-a_{1})}(-z)^{-a_{1}}\,_{3}F_{2}\left(\begin{matrix}a_{1},&1+a_{1}-b_{1},&1+a_{1}-b_{2}\\\
\\\ &1+a_{1}-a_{2},&1+a_{1}-a_{3}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\
+\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(a_{1}-a_{2})\Gamma(a_{3}-a_{2})}{\Gamma(a_{1})\Gamma(a_{3})\Gamma(b_{1}-a_{2})\Gamma(b_{2}-a_{2})}(-z)^{-a_{2}}\,_{3}F_{2}\left(\begin{matrix}a_{2},&1+a_{2}-b_{1},&1+a_{2}-b_{2}\\\
\\\ &1+a_{2}-a_{1},&1+a_{2}-a_{3}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\
+\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(a_{1}-a_{3})\Gamma(a_{2}-a_{3})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(b_{1}-a_{3})\Gamma(b_{2}-a_{1})}(-z)^{-a_{3}}\,_{3}F_{2}\left(\begin{matrix}a_{3},&1+a_{3}-b_{1},&1+a_{3}-b_{2}\\\
\\\ &1+a_{3}-a_{1},&1+a_{3}-a_{2}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)$
(60)

  * •

Analytic continuation of ${}_{4}F_{3}$ around $z=\infty$

$_{4}F_{3}\left(\begin{matrix}a_{1},&a_{2},&a_{3}&a_{4}\\\ \\\
&b_{1},&b_{2}&b_{3}\end{matrix}\Bigg{|}{z}\right)=\\\
\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(b_{3})\Gamma(a_{2}-a_{1})\Gamma(a_{3}-a_{1})\Gamma(a_{4}-a_{1})}{\Gamma(a_{2})\Gamma(a_{3})\Gamma(a_{4})\Gamma(b_{1}-a_{1})\Gamma(b_{2}-a_{1})\Gamma(b_{3}-a_{1})}(-z)^{-a_{1}}\\\
\\\
\,{}_{4}F_{3}\left(\begin{matrix}a_{1},&1+a_{1}-b_{1},&1+a_{1}-b_{2}&1+a_{1}-b_{3}\\\
\\\
&1+a_{1}-a_{2},&1+a_{1}-a_{3}&1+a_{1}-a_{4}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\
+\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(b_{3})\Gamma(a_{1}-a_{2})\Gamma(a_{3}-a_{2})\Gamma(a_{4}-a_{2})}{\Gamma(a_{1})\Gamma(a_{3})\Gamma(a_{4})\Gamma(b_{1}-a_{2})\Gamma(b_{2}-a_{2})\Gamma(b_{3}-a_{2})}(-z)^{-a_{2}}\\\
\\\
\,{}_{4}F_{3}\left(\begin{matrix}a_{2},&1+a_{2}-b_{1},&1+a_{2}-b_{2}&1+a_{2}-b_{3}\\\
\\\
&1+a_{2}-a_{1},&1+a_{2}-a_{3}&1+a_{2}-a_{4}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\
+\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(b_{3})\Gamma(a_{1}-a_{3})\Gamma(a_{2}-a_{3})\Gamma(a_{4}-a_{3})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{4})\Gamma(b_{1}-a_{3})\Gamma(b_{2}-a_{3})\Gamma(b_{3}-a_{3})}(-z)^{-a_{3}}\\\
\\\
\,{}_{4}F_{3}\left(\begin{matrix}a_{3},&1+a_{3}-b_{1},&1+a_{3}-b_{2}&1+a_{3}-b_{3}\\\
\\\
&1+a_{3}-a_{1},&1+a_{3}-a_{2}&1+a_{3}-a_{4}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\
+\frac{\Gamma(b_{1})\Gamma(b_{2})\Gamma(b_{3})\Gamma(a_{1}-a_{4})\Gamma(a_{2}-a_{4})\Gamma(a_{3}-a_{4})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{3})\Gamma(b_{1}-a_{4})\Gamma(b_{2}-a_{4})\Gamma(b_{3}-a_{4})}(-z)^{-a_{4}}\\\
\\\
\,{}_{4}F_{3}\left(\begin{matrix}a_{4},&1+a_{4}-b_{1},&1+a_{4}-b_{2}&1+a_{4}-b_{3}\\\
\\\
&1+a_{4}-a_{1},&1+a_{4}-a_{2}&1+a_{4}-a_{3}\end{matrix}\Bigg{|}{\frac{1}{z}}\right)\\\
\\\ $ (61)

  * •

The analytic continuation around $z=\infty$ of ${}_{p}F_{p-1}(\dots;z)$
function can be found using the Olsson.wl package.

###  6.2 Appendix- B

In this appendix we give an outline of the various methods that are used to
find the ROC of the hypergeometric series [2]. The methods are applicable to
series with more than two variables as well. But the discussion below is
focused on the double variable hypergeometric series and are implemented in
the companion package ROC2.wl of Olsson.wl

  1. 1.

Cancellation of parameters : Cancellation of parameters states that the region
of convergence of a hypergeometric series is independent of the Pochhammer
parameters.
For example, Consider the following Kampé de Fériet function

$S_{1}=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}(b)_{m}(c)_{n}(d)_{n}}{(e)_{m}(f)_{n}(g)_{n}m!n!}x^{m}y^{n}$
(62)

Since the ROC is independent of the Pochhammer parameters, we can choose $d=g$
and the two Pochhammer symbols cancel and the series then effectively becomes
similar to the Appell’s $F_{2}$ function and thus have the ROC: $|x|+|y|<1$

  2. 2.

Horn’s theorem : Horn’s theorem provides a more general way to find the ROC of
a given hypergeometric series for any number of variables. The automatized
implementation of the Horn’s theorem for two variable hypergeometric series is
performed in the package ROC2.wl. We will explain it’s working principle with
the an illustrative example below. Consider the following series

$S_{2}=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}(b)_{m+n}(c)_{m}}{(d)_{2m+n}m!n!}X^{m}Y^{n}$
(63)

Writing the above series in a compact form

$S_{2}=\sum_{m,n=}^{\infty}A_{m,n}X^{m}Y^{n}$ (64)

We then evaluate following two ratios

$\displaystyle f(m,n)$
$\displaystyle=\frac{A_{m+1,n}}{A_{m,n}}=\frac{(a+m+n)(b+m+n)(c+m)}{(d+2m+n)(d+1+2m+n)(1+m)}$
(65) $\displaystyle g(m,n)$
$\displaystyle=\frac{A_{m,n+1}}{A_{m,n}}=\frac{(a+m+n)(b+m+n)}{(d+2m+n)(1+n)}$
(66)

Now we define two more functions as follows

$\displaystyle\Phi(\mu,\nu)$ $\displaystyle=|\lim_{t\to\infty}f(\mu t,\nu
t)|^{-1}$ $\displaystyle\Psi(\mu,\nu)$ $\displaystyle=|\lim_{t\to\infty}g(\mu
t,\nu t)|^{-1}$

From this one can construct following two subsets of $\mathbb{R}_{+}^{2}$

$\displaystyle C$ $\displaystyle=\\{(r,s)\mid 0<r<\Phi(1,0)\wedge
0<s<\Psi(0,1)\\}$ (67) $\displaystyle\doteq K[\Phi(1,0),\Psi(0,1)]$ (68)

and

$Z=\left\\{(r,s)\mid\forall(m,n)\in\mathbb{R}_{+}^{2}:0<r<\Phi(m,n)\vee
0<s<\Psi(m,n)\right\\}$ (69)

The Horn’s theorem for two variable hypergeoemetric functions is then given as
follows

###### Theorem 1

The union of $Z\cap C$ and its projections upon the co-ordinate axes is the
representation in the absolute quadrant $\mathbb{R}_{+}^{2}$ of the region of
convergence in $\mathbb{C}^{2}$ for the series $F$.

For $S_{2}$ we then have

$\displaystyle C$ $\displaystyle=0<r<4\wedge 0<s<1$ $\displaystyle Z$
$\displaystyle=r+s<2s^{1/2}$

Using Horn’s theorem the ROC of the series $S_{2}$ is then given by

$|x|<4\wedge|y|<1\wedge|x|+|y|<2|y|^{1/2}$ (70)

###  6.3 Appendix C

In this appendix we briefly describe the Mathematica [22] package HornH1H5.wl
for the ease of the reader. Since it is not possible to present all the ACs of
the Horn $H_{1}$ and $H_{5}$ function due to their lengthy expressions so we
provide this package as an ancillary file. The package can be downloaded from
the link below.

https://github.com/souvik5151/Horn_H1_H5.git

Once the file is downloaded, it can be called, after setting the correct path,
as

In[]:= <<HornH1H5.wl

This package can be used to access all the ACs as well as their corresponding
ROCs. We have included the two commands for each of the function $H_{1}$ and
$H_{5}$ in the package. These are the following

  1. 1.

H1expose and H5expose : These commands take an integer ( less than the number
of ACs ) and gives the output as a list of two elements containing the ROC and
the expression of the AC, labelled by that number, of $H_{1}$ and $H_{5}$
respectively. There are 22 ACs of $H_{1}$ and 20 ACs of $H_{5}$.

  2. 2.

H1ROC and H5ROC : These commands plots the ROC of a particular AC for real
values of $x$ and $y$, which is specified by a number given by the user, along
with a user specified point $(x,y)$. This command can be used when the user
wants to determine if a certain point lie inside the ROC of an AC or not.

More information about these commands are available to the user via the
Information command of Mathematica after loading the package.

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