# The asymptotic expansion of Krätzel’s integral and an integral related to an
extension of the Whittaker function

R. B. Paris
Division of Computing and Mathematics,
Abertay University, Dundee DD1 1HG, UK

###### Abstract

We consider the asymptotic expansion of Krätzel’s integral

$F_{p,\nu}(x)=\int_{0}^{\infty}t^{\nu-1}e^{-t^{p}-x/t}\,dt\qquad(|\arg\,x|<\pi/2),$

for $p>0$ as $|x|\to\infty$ in the sector $|\arg\,x|<\pi/2$ employing the
method of steepest descents. An alternative derivation of this expansion is
given using a Mellin-Barnes integral approach. The cases $p<0$, $\Re(\nu)<0$
and when $x$ and $\nu$ ($p>0$) are both large are also considered.

A second section discusses the asymptotic expansion of an integral involving a
modified Bessel function that has recently been introduced as an extension of
the Whittaker function $M_{\kappa,\mu}(z)$. Numerical examples are provided to
illustrate the accuracy of the various expansions obtained.

Mathematics subject classification (2010): 30E15, 33E20, 33C15, 34E05, 41A60

Keywords: Krätzel’s integral, asymptotic expansions, steepest descents,
Mellin-Barnes integral, modified Bessel function, extended Whittaker function

1\. Introduction

The first part of this paper is concerned with the asymptotic expansion of
Krätzel’s integral defined by111In [4] this integral was denoted by
$Z_{p}^{\nu}(x)$.

$F_{p,\nu}(x)=\int_{0}^{\infty}t^{\nu-1}\exp\,\biggl{(}-t^{p}-\frac{x}{t}\biggr{)}\,dt\qquad(|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi),$
(1.1)

where $p\in{\bf R}$, $\nu\in{\bf C}$, with $\Re(\nu)<0$ when $p\leq 0$. For
$p\geq 1$, the function $F_{p,\nu}(x)$ was introduced by Krätzel [5] as a
kernel of the integral transform

$\int_{0}^{\infty}F_{p,\nu}(xt)\phi(t)\,dt\qquad(x>0).$

He derived the inversion formula for this transform and constructed its
convolution to give applications to the solution of some ordinary differential
equations. Further investigations of $F_{p,\nu}(x)$ were carried out by Kilbas
et al. [4] who established its connection with the Wright function
${}_{1}\Psi_{0}(a,b;z)$ defined in Section 3.

Particular cases of (1.1) are: when $p=1$, $x=z^{2}/4$

$F_{1,\nu}(z^{2}/4)=2(\mbox{${\textstyle\frac{1}{2}}$}z)^{\nu}K_{\nu}(z),$

where $K_{\nu}(z)$ is the modified Bessel function (or Macdonald function);
and when $p=0$ and $p=-1$

$F_{0,\nu}(x)=x^{\nu}\Gamma(-\nu),\qquad
F_{-1,\nu}(x)=(1+x)^{\nu}\Gamma(-\nu)\qquad(\Re(\nu)<0).$ (1.2)

We determine the asymptotic expansion of (1.1) when $p>0$ for $|x|\to\infty$
in the sector $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$ by application
of the method of steepest descents, with the coefficients in the expansion
determined by Lagrange inversion. We present an alternative derivation of the
expansion when $p>0$ using a Mellin-Barnes integral representation of
$F_{p,\nu}(x)$ given in [4]. The case when $p<0$ is discussed in Section 4. We
also consider the expansion in the case when the parameter $\nu$ is large of
$O(x^{p/(p+1)})$ in Section 5.

In the second part, we consider an integral also involving the modified Bessel
function $K_{\nu}(z)$ that arose in an extension of the Whittaker function
$M_{\kappa,\mu}(z)$ discussed in [2]. The integral in question is

$I(x)=\sqrt{\frac{2p}{\pi}}\int_{0}^{1}t^{a-\frac{1}{2}}(1-t)^{b-\frac{1}{2}}e^{-xt}K_{\nu}\biggl{(}\frac{p}{t(1-t)}\biggr{)}\,dt\qquad(\nu\geq-\mbox{${\textstyle\frac{1}{2}}$},\
p>0).$

The leading approximation of $I(x)$ for large $x>0$ was obtained by
application of the saddle-point method in [2, §3]. Here we determine its
asymptotic expansion valid for $|x|\to\infty$ in the sector
$|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$ employing a different
approach.

2\. The expansion of $F_{p,\nu}(x)$ when $p>0$

Let us consider first the case $p>0$ in (1.1). We make the change of variable
$t=\tau(x/p)^{1/(p+1)}$ to find

$F_{p,\nu}(x)=(x/p)^{\nu/(p+1)}\int_{0}^{\infty}\tau^{\nu-1}e^{-X\psi(\tau)}\,d\tau,$
(2.1)

where

$\psi(\tau):=\frac{\tau^{p}}{p}+\frac{1}{\tau},\qquad
X:=p^{1/(p+1)}x^{p/(p+1)}.$ (2.2)

The integral in (2.1) converges at both endpoints provided $\Re(X)>0$; that
is, when $|\arg\,x|<(p+1)\pi/(2p)$ and so represents the analytical
continuation of (1.1) into this wider sector.

Saddle points of the exponential factor occur when $\psi^{\prime}(\tau)=0$;
that is, when $\tau^{p-1}-1/\tau^{2}=0$ whence

$\tau_{sk}=e^{2\pi ik/(p+1)}\qquad(k=0,\pm 1,\pm 2,\ldots).$

The saddle points are consequently situated on the unit circle in the
$\tau$-plane, with the saddles corresponding to higher $k$ values being
situated on different Riemann surfaces. The only contributory saddle for
$F_{p,\nu}(x)$ is found to be that corresponding to $k=0$; see Fig. 1 for
typical examples of the steepest descent path from the origin passing to
infinity in the direction $\arg\,\tau=-\theta/(p+1)$, where $\theta=\arg\,x$.

($a$) ($b$)

Figure 1: The path of steepest descent from the origin passing over the saddle
at $\tau=1$ when $p=2$: ($a$) $\arg\,X=0$ and ($b$) $\arg\,X=\pi/3$. The
arrows denote the direction of integration and the dots represent the saddle
points situated at $\tau=1,e^{\pm 2\pi i/3}$.

Noting that $\psi(1)=(p+1)/p$, we introduce the new variable $w$ by

$\frac{1}{2}w^{2}=\psi(\tau)-\psi(1)=\frac{\psi^{\prime\prime}(1)}{2!}(\tau-1)^{2}+\frac{\psi^{\prime\prime\prime}(1)}{3!}(\tau-1)^{3}+\cdots$

so that

$F_{p,\nu}(x)=(x/p)^{\nu/(p+1)}e^{-X\psi(1)}\int_{-\infty}^{\infty}e^{-X\tau^{2}/2}\tau^{\nu-1}\frac{d\tau}{dw}\,dw.$

Inversion of the above expansion using the InverseSeries command in
Mathematica (essentially Lagrange inversion) yields

$\tau=1+\frac{1}{(p+1)^{1/2}}\sum_{k=1}^{\infty}A_{k}w^{k},$ (2.3)

where the first few coefficients $A_{k}$ are

$A_{1}=1,\quad A_{2}=\frac{4-p}{6(p+1)^{1/2}},\quad
A_{2}=\frac{26-19p+2p^{2}}{72(p+1)},$
$A_{3}=\frac{2}{135(p+1)^{3/2}}(46-66p+21p^{2}-p^{3}),$
$A_{4}=\frac{1}{3456(p+1)^{2}}(1252-2956p+1881p^{2}-316p^{3}+4p^{4}).$

Then we have

$\frac{d\tau}{dw}=\frac{1}{(p+1)^{1/2}}\sum_{k=1}^{\infty}kA_{k}w^{k-1}.$
(2.4)

Combining (2.3) and (2.4), we obtain

$\tau^{\nu-1}\frac{d\tau}{dw}\stackrel{{\scriptstyle
e}}{{=}}\frac{1}{(p+1)^{1/2}}\sum_{k=0}^{\infty}B_{k}w^{2k},$

where $\stackrel{{\scriptstyle e}}{{=}}$ denotes the inclusion of only the
even powers of $w$ (since the odd powers will not contribute to the integral).
The first few coefficients $B_{k}$ are given by:

$B_{0}=1,\quad B_{1}=\frac{1}{24(p+1)}(2+12\nu^{2}-12\nu(p-1)-7p+2p^{2}),$
$B_{2}=\frac{1}{3456(p+1)^{2}}\biggl{\\{}4+144\nu^{4}-480\nu^{3}(p-1)-172p+417p^{2}-172p^{3}+4p^{4}$
$\hskip
142.26378pt+120\nu^{2}(4-11p+4p^{2})-24\nu(-6+41p(1-p)+6p^{3})\biggr{\\}}.$

It is evident that the rapidly increasing complexity of the higher
coefficients prevents their explicit representation. However, when dealing
with specific values of the parameters $p$ and $\nu$, it is quite feasible to
generate numerically many coefficients $B_{k}$ using the numerical inversion
procedure described above.

Then

$F_{p,\nu}(x)\sim(x/p)^{\nu/(p+1)}e^{-X\psi(1)}\sum_{k=0}^{\infty}B_{k}\int_{-\infty}^{\infty}e^{-Xw^{2}/2}w^{2k}dw$
$=\sqrt{\frac{2\pi}{(p+1)X}}\,\frac{e^{-X\psi(1)}}{(p/x)^{\nu/(p+1)}}\sum_{k=0}^{\infty}\frac{(\mbox{${\textstyle\frac{1}{2}}$})_{k}B_{k}}{(X/2)^{k}}\hskip
8.5359pt$ (2.5)

as $x\to+\infty$, where $X$ is defined in (2.2) and
$(a)_{k}=\Gamma(a+k)/\Gamma(a)$ is Pochhammer’s symbol. We observe that the
argument of the exponential factor is

$X\psi(1)=(p+1)\,(x/p)^{p/(p+1)}.$

The contributory saddle continues to be $\tau_{s0}=1$ when $x$ is complex
satisfying $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$; see Fig. 1(b).
Hence the expansion in (2.5) continues to hold for $|x|\to\infty$ in the
sector $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$.

To illustrate the accuracy of the expansion (2.5) we present the cases $p=2$,
$\nu=\mbox{${\textstyle\frac{1}{2}}$}$ and $p=3$,
$\nu=\mbox{${\textstyle\frac{3}{2}}$}$. In Table 1 we show the values of the
coefficients $B_{k}$ for $k\leq 6$ obtained by numerical inversion. In Table 2
we give the absolute relative error in the computation of $F_{p,\nu}(x)$ using
(2.5) for different truncation index $k$. The exact value of $F_{p,\nu}(x)$
was determined by high-precision evaluation of the integral in (1.1).

Table 1: The coefficients $B_{k}$ (with $B_{0}=1$) for $1\leq k\leq 6$. $k$ | $p=0.75,\ \nu=1.50$ | $p=1.50,\ \nu=0.50$
---|---|---
1 | $+6.99404762\times 10^{-1}$ | $-6.66666667\times 10^{-2}$
2 | $+4.53883811\times 10^{-2}$ | $+9.07407407\times 10^{-3}$
3 | $-1.50268380\times 10^{-3}$ | $-1.24847737\times 10^{-3}$
4 | $+9.39269762\times 10^{-5}$ | $+1.63694787\times 10^{-4}$
5 | $-6.82774001\times 10^{-6}$ | $-1.97786516\times 10^{-5}$
6 | $+5.11051949\times 10^{-7}$ | $+2.08358000\times 10^{-6}$
Table 2: Values of the absolute relative error in $F_{p,\nu}(x)$ using (2.5) for different truncation index $k$ when $x=30$. The final row gives the value of $F_{p,\nu}(x)$. $k$ | $p=0.75,\ \nu=1.50$ | $p=1.50,\ \nu=0.50$
---|---|---
0 | $1.679\times 10^{-2}$ | $7.106\times 10^{-3}$
1 | $8.182\times 10^{-3}$ | $3.115\times 10^{-4}$
2 | $3.478\times 10^{-4}$ | $2.313\times 10^{-5}$
3 | $3.966\times 10^{-5}$ | $2.305\times 10^{-6}$
4 | $6.844\times 10^{-6}$ | $2.744\times 10^{-7}$
5 | $1.504\times 10^{-6}$ | $3.542\times 10^{-8}$
$F_{p,\nu}(x)$ | $5.70691385\times 10^{-3}$ | $2.67529872\times 10^{-7}$

3\. An alternative derivation of the expansion for $F_{p,\nu}(x)$ when $p>0$

In Kilbas et al. [4] it was shown that

$F_{p,\nu}(x)=\frac{1}{2\pi ip}\int_{c-\infty i}^{c+\infty
i}\Gamma(s)\Gamma\biggl{(}\frac{s+\nu}{p}\biggr{)}x^{-s}ds\qquad
c>\max\\{0,-\Re(\nu)\\}$ (3.1)

valid in the sector $|\arg\,x|<(p+1)\pi/(2p)$. The integration path lies to
the right of the poles of $\Gamma(s)$ and $\Gamma((s+\nu)/p)$ situated at
$s=-k$ and $s=-kp-\nu$ ($k=0,1,2,\ldots$), respectively. This result can be
easily established by making use of the Cahen-Mellin integral (see, for
example, [9, p. 89])

$e^{-z}=\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty
i}\Gamma(s)z^{-s}ds\qquad(c>0;\
|\arg\,z|<\mbox{${\textstyle\frac{1}{2}}$}\pi)$ (3.2)

with $z=x/t$ followed by substitution in (1.1) and evaluation of the resulting
integral over $t$ as a gamma function.

Displacement of the integration path to the left over the poles produces
(provided $kp+\nu$ does not take on non-negative integer values) [4]

$F_{p,\nu}(x)=\frac{1}{p}\sum_{k=0}^{\infty}\Gamma\biggl{(}\frac{\nu-k}{p}\biggr{)}\,\frac{(-x)^{k}}{k!}+x^{\nu}\sum_{k=0}^{\infty}\Gamma(-\nu-
kp)\,\frac{(-x^{p})^{k}}{k!}$ $\ \
=\frac{1}{p}{}_{1}\Psi_{0}(-1/p,\nu/p;-x)+x^{\nu}{}_{1}\Psi_{0}(-p,-\nu:-x^{p}),$
(3.3)

where the Wright function ${}_{1}\Psi_{0}(a,b;z)$ is defined by

${}_{1}\Psi_{0}(a,b;z):=\sum_{k=0}^{\infty}\Gamma(ak+b)\,\frac{z^{k}}{k!}\qquad(0<a<1).$

When $kp+\nu$ assumes a non-negative integer value, double poles arise with
the result that logarithmic terms will be present.

To determine the asymptotic expansion of the integral in (3.1) we may displace
the integration path as far to the right as we please, since there are no
poles of the integrand on the right of this path. On the displaced path, which
we denote by $L$, it follows that $|s|$ is everywhere large on $L$. If we
define the parameters

$\kappa=\frac{p+1}{p},\quad
h=p^{1/p},\quad\vartheta=\frac{\nu}{p}-\frac{1}{2},\quad{\cal
A}_{0}=(2\pi)^{-1/2}\kappa^{\frac{1}{2}-\vartheta}p^{-\vartheta},$

we have the inverse factorial expansion [9, p. 39, Lemma 2.2]

$\Gamma(s)\Gamma\biggl{(}\frac{s+\nu}{p}\biggr{)}=2\pi{\cal
A}_{0}(h\kappa^{\kappa})^{-s}\biggl{\\{}\sum_{j=0}^{M-1}(-)^{j}c_{j}\Gamma(\kappa
s+\vartheta-j)+\rho_{M}(s)\Gamma(\kappa s+\vartheta-M)\biggr{\\}},$ (3.4)

where $M$ is a positive integer and $\rho_{M}(s)=O(1)$ as $|s|\to\infty$ in
$|\arg\,s|<\pi$. The coefficient $c_{0}=1$ and an algorithm for the
determination of the higher coefficients $c_{j}$ ($j\geq 1$) is given in [9,
§2.2.4], [8, pp. 321–323].

Substitution of (3.4) into (3.1) taken over the displaced integration path $L$
then yields

$F_{p,\nu}(x)=\frac{2\pi{\cal
A}_{0}}{p}\biggl{\\{}\sum_{j=0}^{M-1}(-)^{j}c_{j}\,\frac{1}{2\pi
i}\int_{L}(\kappa X)^{-\kappa s}\Gamma(\kappa
s+\vartheta-j)\,ds+R_{M}(x)\biggr{\\}},$

where $X=(hx)^{1/\kappa}=p^{1/(p+1)}x^{p/(p+1)}$ (see (2.2)) and the remainder
term $R_{M}(x)$ is

$R_{M}(x)=\frac{1}{2\pi i}\int_{L}\rho_{M}(s)(\kappa X)^{-\kappa
s}\Gamma(\kappa s+\vartheta-M)\,ds.$

Application of Lemma 2.7 in [9, p. 71] shows that

$R_{M}(x)=O(X^{\vartheta-M}e^{-\kappa X})\qquad(|X|\to\infty,\
|\arg\,X|<\mbox{${\textstyle\frac{1}{2}}$}\pi).$

Hence, upon use of the Cahen-Mellin integral in (3.2), we obtain the expansion

$F_{p,\nu}(x)=\frac{2\pi{\cal A}_{0}}{\kappa p}\,X^{\vartheta}e^{-\kappa
X}\biggl{\\{}\sum_{j=0}^{M-1}(-)^{j}c_{j}(\kappa X)^{-j}+O(X^{-M})\biggr{\\}}$
(3.5)

as $|X|\to\infty$ in the sector
$|\arg\,X|<\mbox{${\textstyle\frac{1}{2}}$}\pi$. We note that this sector
includes the sector $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$.

Routine algebra shows that the leading term of this expansion agrees with the
leading term of (2.5). Consequently we must have the coefficients $c_{j}$
related to the $B_{j}$ in (2.5) given by

$c_{j}=(-)^{j}(2\kappa)^{j}(\mbox{${\textstyle\frac{1}{2}}$})_{j}B_{j}.$

4\. The expansion for $F_{p,\nu}(x)$ when $p<0$

When $p<0$ the expansions of $F_{p,\nu}(x)$ are different according as $-1\leq
p<0$ and $p\leq-1$; the situations pertaining to $p=0$ and $p=-1$ being given
in (1.2). In both ranges of the parameter $p$ we require $\Re(\nu)<0$ for
convergence of the integral in (1.1) at the upper limit.

From [4, (3.13)], we have when $p<0$

$F_{p,\nu}(x)=-\frac{1}{2\pi ip}\int_{c-\infty i}^{c+\infty
i}\Gamma(s)\Gamma\biggl{(}\frac{s+\nu}{p}\biggr{)}x^{-s}\,ds\qquad
0<c<-\Re(\nu),$

where the integration path now separates the poles of $\Gamma(s)$ at $s=-k$
from those of $\Gamma((s+\nu)/p)$ at $s=-kp-\nu$, ($k=0,1,2,\ldots$). When
$-1\leq p<0$, displacement of the path to the right over the poles at
$s=-kp-\nu$ produces [4, Theorem 4.2]

$F_{p,\nu}(x)=x^{\nu}\sum_{k=0}^{\infty}\frac{\Gamma(-kp-\nu)}{k!}\,(-x^{p})^{k}\qquad(\Re(\nu)<0,\
-1\leq p<0).$ (4.1)

This convergent series also yields the asymptotic expansion for $|x|\to\infty$
valid in $|\arg\,x|<(p+1)\pi/(2p)$. Hence we have

$F_{p,\nu}(x)=x^{\nu}\biggl{\\{}\Gamma(-\nu)-\frac{\Gamma(-p-\nu)}{x^{-p}}+O(1/x^{-2p})\biggr{\\}}\qquad(x\to\infty).$

When $p\leq-1$, displacement of the path to the left over the poles at $s=-k$
produces [4, Theorem 4.3]

$F_{p,\nu}(x)=-\frac{1}{p}\sum_{k=0}^{\infty}\Gamma\biggl{(}\frac{\nu-k}{p}\biggr{)}\,\frac{(-x)^{k}}{k!}\qquad(\Re(\nu)<0,\
p\leq-1).$ (4.2)

The asymptotic expansion in this case is obtained by displacing the
integration path to the right over the poles at $s=-kp-\nu$ to yield

$F_{p,\nu}(x)\sim
x^{\nu}\sum_{k=0}^{\infty}\Gamma(-kp-\nu)\,\frac{(-x^{p})^{k}}{k!}\qquad(\Re(\nu)<0,\
p\leq-1)$ (4.3)

as $|x|\to\infty$ in $|\arg\,x|<(p+1)\pi/(2p)$.

It is easily verified that when $p=-1$ the expansions (4.1) and (4.2) both
reduce to

$F_{-1,\nu}(x)=(1+x)^{\nu}\Gamma(-\nu)$

as stated in (1.2).

5\. The expansion of $F_{p,\nu}(x)$ when $\nu$ and $x$ are large

Let $\nu=1+aX$, where $a>0$ is fixed and $X=p^{1/(p+1)}x^{p/(p+1)}$ with
$p>0$. Then from (2.1) we find

$F_{p,\nu}(x)=(x/p)^{\nu/(p+1)}\int_{0}^{\infty}\tau^{aX}e^{-X(\tau^{p}/p+\tau^{-1})}d\tau=(x/p)^{\nu/(p+1)}\int_{0}^{\infty}e^{-X\phi(\tau)}d\tau,$
(5.1)

where

$\phi(\tau)=\frac{\tau^{p}}{p}+\frac{1}{\tau}-a\log\,\tau.$

Saddle points of the exponential factor are given by solutions of the equation

$\tau^{p+1}=1+a\tau.$ (5.2)

Examination of the steepest paths shows that the path from the origin through
the contributory saddle $\tau_{s}>1$ on the positive real axis is similar to
that shown in Fig. 1. Making the change of variable

$\frac{1}{2}w^{2}=\phi(\tau)-\phi(\tau_{s}),$

we find upon inversion as described in Section 2, followed by differentiation,

$\frac{d\tau}{dw}\stackrel{{\scriptstyle
e}}{{=}}\frac{1}{\sqrt{\phi_{s}^{\prime\prime}}}\,\sum_{k\geq 0}C_{k}w^{2k},$

where we use the abbreviation $\phi_{s}=\phi(\tau_{s})$ and

$C_{0}=1,\qquad C_{1}=\frac{1}{24\phi_{s}^{\prime\prime
3}}(5\phi_{s}^{\prime\prime\prime 2}-3\phi_{s}^{\prime\prime}\phi_{s}^{iv}),$
$C_{2}=\frac{1}{3456\phi_{s}^{\prime\prime 6}}(385\phi_{s}^{\prime\prime\prime
4}-630\phi_{s}^{\prime\prime}\phi_{s}^{\prime\prime\prime
2}\phi_{s}^{iv}+105\phi_{s}^{\prime\prime
2}(\phi_{s}^{iv})^{2}+168\phi_{s}^{\prime\prime
2}\phi_{s}^{\prime\prime\prime}\phi_{s}^{v}-24\phi_{s}^{\prime\prime
3}\phi_{s}^{vi}).$

Then

$F_{p,\nu}(x)=(x/p)^{\nu/(p+1)}\,\frac{e^{-X\phi(\tau_{s})}}{\sqrt{\phi_{s}^{\prime\prime}}}\int_{-\infty}^{\infty}e^{-Xw^{2}/2}\,\frac{d\tau}{dw}\,dw$
$\sim(x/p)^{\nu/(p+1)}\,e^{-X\phi(\tau_{s})}\,\sqrt{\frac{2\pi}{X\phi_{s}^{\prime\prime}}}\sum_{k\geq
0}\frac{(\mbox{${\textstyle\frac{1}{2}}$})_{k}C_{k}}{(X/2)^{k}}$

as $x\to\infty$.

Upon noting that

$\phi(\tau_{s})=\frac{p+1}{p\tau_{s}}+\frac{a}{p}-a\log\,\tau_{s},\qquad\phi_{s}^{\prime\prime}=\frac{1}{\tau_{s}^{2}}\biggl{(}\frac{p+1}{\tau_{s}}+ap\biggr{)},$

where use has been made of (5.2), we finally obtain the expansion

$F_{p,\nu}(x)\sim\sqrt{\frac{2\pi}{X\phi_{s}^{\prime\prime}}}\,\frac{e^{-\kappa
X/\tau_{s}-aX/p}}{(p/x)^{\nu/(p+1)}}\,\tau_{s}^{aX}\sum_{k\geq
0}\frac{(\mbox{${\textstyle\frac{1}{2}}$})_{k}C_{k}}{(X/2)^{k}}$ (5.3)

as $x\to\infty$ when $\nu=1+aX$, $X=p^{1/(p+1)}x^{p/(p+1)}$ and
$\kappa=(p+1)/p$. It is found that the steepest descent path when $x$ is
complex is very similar to that shown in Fig. 1(b), with the real positive
saddle $\tau_{s}$ still being the only contributory saddle when
$|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$. Hence the expansion (5.3)
holds for complex $x$ when $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$.

Since, in general, the saddle $\tau_{s}$ has to be obtained by numerical
solution of (5.2) it is reasonable to determine the coefficients $C_{k}$
numerically using the InverseSeries command in Mathematica. As an example, we
display the coefficients in Table 3 for two cases of $(p,a)$ and in Table 4
the corresponding values of the absolute relative error in the expansion (5.3)
for complex $x$.

Table 3: The coefficients $C_{k}$ (with $C_{0}=1$) for $1\leq k\leq 5$. $k$ | $p=2,\ a=1$ | $p=3,\ a=0.50$
---|---|---
1 | $-5.12469537\times 10^{-2}$ | $-0.09820959\times 10^{-2}$
2 | $+4.75849832\times 10^{-3}$ | $+5.38592491\times 10^{-3}$
3 | $-6.64820214\times 10^{-5}$ | $+1.51551186\times 10^{-3}$
4 | $-4.28119981\times 10^{-5}$ | $-4.26687479\times 10^{-4}$
5 | $+5.50086099\times 10^{-6}$ | $+3.10584484\times 10^{-5}$
Table 4: Values of the absolute relative error in $F_{p,\nu}(x)$ when $\nu=1+aX$ using (5.3) when $x=30e^{i\theta}$ for different $\theta$ and truncation index $k=3$. | $p=2,\ a=1$ | $p=3,\ a=0.50$
---|---|---
$\theta/\pi$ | $\tau_{s}=1.32471796$ | $\tau_{s}=1.11734904$
0 | $1.870\times 10^{-7}$ | $5.319\times 10^{-7}$
0.10 | $1.874\times 10^{-7}$ | $5.326\times 10^{-7}$
0.20 | $1.886\times 10^{-7}$ | $5.349\times 10^{-7}$
0.30 | $1.905\times 10^{-7}$ | $5.385\times 10^{-7}$
0.40 | $1.931\times 10^{-7}$ | $5.431\times 10^{-7}$
0.45 | $1.947\times 10^{-7}$ | $5.457\times 10^{-7}$

6\. An integral connected with an extended Whittaker function

In a recent paper [2], an extension of the Whittaker function was introduced
by

$M_{\kappa,\mu}^{(p,\nu)}(z)=\frac{x^{\mu+\frac{1}{2}}e^{x/2}}{B(\mu-\kappa+\frac{1}{2},\mu+\kappa+\frac{1}{2})}\sqrt{\frac{2p}{\pi}}\int_{0}^{1}t^{\mu+\kappa-1}(1-t)^{\mu-\kappa-1}e^{zt}K_{\nu+\frac{1}{2}}\biggl{(}\frac{p}{t(1-t)}\biggr{)}\,dt,$

where $K_{\nu}(z)$ is the modified Bessel function and
$B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ is the beta function. This
representation is valid for $p\geq 0$, $\nu\geq 0$ and $2\mu\neq-1,-2,\ldots\
$. In the case $p=\nu=0$, the extended function reduces to the usual Whittaker
function $M_{\kappa,\mu}(x)$ given by [7, p. 337]

$M_{\kappa,\mu}(z)=\frac{x^{\mu+\frac{1}{2}}e^{-z/2}}{B(\mu-\kappa+\frac{1}{2},\mu+\kappa+\frac{1}{2})}\int_{0}^{1}t^{\mu+\kappa-\frac{1}{2}}(1-t)^{\mu-\kappa-\frac{1}{2}}e^{zt}dt.$

Our aim is to determine the asymptotic expansion of the related integral

$I(a,b;z)=\sqrt{\frac{2p}{\pi}}\int_{0}^{1}t^{a-\frac{1}{2}}(1-t)^{b-\frac{1}{2}}e^{zt}K_{\nu}\biggl{(}\frac{p}{t(1-t)}\biggr{)}\,dt\qquad(\nu\geq-\mbox{${\textstyle\frac{1}{2}}$},\
p>0),$ (6.1)

valid for arbitrary parameters $a$ and $b$, as $|z|\to\infty$. This integral
corresponds to that appearing in the definition of
$M_{\kappa,\mu}^{(p,\nu)}(x)$ when
$a=\mu+\kappa-\mbox{${\textstyle\frac{1}{2}}$}$,
$b=\mu-\kappa-\mbox{${\textstyle\frac{1}{2}}$}$ and
$\nu\to\nu+\mbox{${\textstyle\frac{1}{2}}$}$. By making the change of variable
$t\to 1-t$, it is easily seen that $I(a,b;z)$ satisfies the following Kummer-
type transformation

$I(a,b;z)=e^{z}I(b,a;-z).$ (6.2)

We first consider the expansion of $I(a,b;z)$ for large $z$ in the sector
$|\arg(-z)|<\mbox{${\textstyle\frac{1}{2}}$}\pi$. We set $z=-x$ and note that
for large $x$ the dominant behaviour of the integrand results from a
neighbourhood of $t=0$. From [7, (10.40.2)] we have the expansion

$K_{\nu}\biggl{(}\frac{p}{t(1-t)}\biggr{)}\sim\sqrt{\frac{\pi}{2p}}\exp\,\biggl{[}-\frac{p}{t}-\frac{p}{1-t}\biggr{]}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{p^{k}}\,(t(1-t))^{k+\frac{1}{2}}$

as $t\to 0$, where $a_{0}(\nu)=1$ and

$a_{k}(\nu)=\prod_{r=1}^{k}\frac{4\nu^{2}-(2r-1)^{2}}{k!\
2^{3k}}=(-1)^{k}\frac{(\mbox{${\textstyle\frac{1}{2}}$}+\nu)_{k}(\mbox{${\textstyle\frac{1}{2}}$}-\nu)_{k}}{k!2^{k}}\quad(k\geq
1).$

Substitution of this expansion into the integral (6.1) then leads to

$I(a,b;-x)\sim\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{p^{k}}\int_{0}^{1}t^{a+k}(1-t)^{b+k}e^{-xt-p/t}\,e^{-p/(1-t)}dt$
$\hskip
31.2982pt=\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{p^{k}x^{a+k+1}}\int_{0}^{x}\tau^{a+k}e^{-(\tau+px/\tau)}g_{k}(\tau/x)\,d\tau,$

where

$g_{k}(w):=(1-w)^{b+k}e^{-p/(1-w)}=e^{-p}(1-w)^{b+k}e^{-pw/(1-w)}.$ (6.3)

We can expand $g_{k}(w)$ about $w=0$ to obtain

$g_{k}(w)=e^{-p}\sum_{r=0}^{\infty}\frac{(-1)^{r}c_{r}(k)}{r!}w^{r}\qquad(|w|<1),$

where the first few coefficients $c_{r}(k)$ are given by

$c_{0}(k)=1,\qquad c_{1}(k)=p+\beta,$ $c_{2}(k)=(p+\beta)^{2}-(p+\beta)-p,$
$c_{3}(k)=(p+\beta)^{3}-6(p+\beta)^{2}+(3b+2)(p+\beta)+4p,$ (6.4)

with $\beta:=b+k$. Then

$I(a,b;-x)\sim
e^{-p}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{p^{k}x^{a+k+1}}\sum_{r=0}^{\infty}\frac{(-1)^{r}c_{r}(k)}{r!\,x^{r}}\int_{0}^{x}\tau^{a+k+r}e^{-(\tau+px/\tau)}d\tau.$

We now extend the upper limit $x$ to $\infty$, thereby introducing an error of
$O(e^{-x})$ when $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$. Employing
the integral representation of the modified Bessel function [7, (10.32.10)]

$2(\mbox{${\textstyle\frac{1}{2}}$}z)^{-\nu}K_{\nu}(z)=\int_{0}^{\infty}e^{-(\tau+z^{2}/(4\tau))}\,\frac{d\tau}{\tau^{\nu+1}}\qquad(|\arg\,z|<\mbox{${\textstyle\frac{1}{4}}$}\pi)$

and the fact that $K_{-\nu}(z)=K_{\nu}(z)$, we find (with $\omega:=a+1+k+r$)

$I(a,b;-x)\sim
2e^{-p}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{p^{k}x^{a+k+1}}\sum_{r=0}^{\infty}\frac{(-1)^{r}c_{r}(k)}{r!\,x^{r}}\,(px)^{\mu/2}K_{\omega}(2\sqrt{px})$
$\hskip
34.14322pt=2e^{-p}\biggl{(}\frac{p}{x}\biggr{)}^{(a+1)/2}\sum_{k=0}^{\infty}a_{k}(\nu)\sum_{r=0}^{\infty}\frac{(-1)^{r}c_{r}(k)p^{r}}{r!\,(px)^{(k+r)/2}}\,K_{\omega}(2\sqrt{px}).$
(6.5)

Table 5: The absolute relative error in the computation of $I(a,b;-x)$ from (6.1) for different truncation index $r$ in the asymptotic expansion (6.6) when $a=b=p=1$, $\nu=4/3$. $r$ | $x=100$ | $x=200$ | $x=500$
---|---|---|---
0 | $1.912\times 10^{-1}$ | $1.181\times 10^{-1}$ | $6.658\times 10^{-2}$
1 | $2.489\times 10^{-2}$ | $1.147\times 10^{-2}$ | $4.267\times 10^{-3}$
2 | $5.124\times 10^{-3}$ | $1.484\times 10^{-3}$ | $3.167\times 10^{-4}$
3 | $1.732\times 10^{-4}$ | $3.503\times 10^{-5}$ | $4.647\times 10^{-6}$
4 | $4.387\times 10^{-5}$ | $5.780\times 10^{-6}$ | $4.499\times 10^{-7}$
5 | $1.236\times 10^{-5}$ | $1.065\times 10^{-6}$ | $4.876\times 10^{-8}$
6 | $4.694\times 10^{-6}$ | $2.620\times 10^{-7}$ | $6.977\times 10^{-9}$

The double series can be summed “diagonally” (cf. [10, p. 58]) to produce the
final result

$I(a,b;-x)\sim
2e^{-p}\biggl{(}\frac{p}{x}\biggr{)}^{(a+1)/2}\sum_{r=0}^{\infty}\frac{D_{r}K_{a+1+r}(2\sqrt{px})}{(px)^{r/2}}$
(6.6)

as $|x|\to\infty$ in $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$, where
$D_{0}=1$ and

$D_{r}=\sum_{\ell=0}^{r}\frac{(-1)^{\ell}p^{\ell}}{\ell!}\,a_{r-\ell}(\nu)c_{\ell}(r-\ell)\qquad(r\geq
1).$

The expansion valid in $|\arg\,z|<\mbox{${\textstyle\frac{1}{2}}$}\pi$
immediately follows from (6.6) and the Kummer-type transformation in (6.2).

The leading asymptotic form is consequently

$I(a,b;-x)\sim
2e^{-p}\biggl{(}\frac{p}{x}\biggr{)}^{(a+1)/2}K_{a+1}(2\sqrt{px})\sim\frac{\sqrt{\pi}p^{a/2+1/4}}{x^{a/2+3/4}}\,e^{-p-2\sqrt{px}}$
(6.7)

as $|x|\to\infty$ in $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$. A result
equivalent to the above was obtained by the saddle-point method in [2, §3].

In Table 5 we show an example of the values of the absolute relative error in
the computation of $I(a,b;-x)$ from (6.1) and (6.6) for different $x$ and
truncation index $r$ in the asymptotic sum. The calculation of the
coefficients $c_{r}(k)$ in (6.3) can be carried out numerically when specific
values of the parameters are known using the InverseSeries command in
Mathematica.

To conclude, we note that an extended confluent hypergeometric function was
introduced in [1] that involved the integral

$J(a,b;z)=\int_{0}^{1}t^{a}(1-t)^{b}e^{zt}\exp\biggl{(}\frac{p}{t(1-t)}\biggr{)}\,dt\qquad(p>0).$

The same arguments employed above produce the expansion

$J(a,b;-x)\sim
2e^{-p}\biggl{(}\frac{p}{x}\biggr{)}^{(a+1)/2}\sum_{r=0}^{\infty}\frac{(-1)^{r}p^{r}c_{r}(0)}{r!(px)^{r/2}}\,K_{a+1+r}(2\sqrt{px})$
(6.8)

as $|x|\to\infty$ in $|\arg\,x|<\mbox{${\textstyle\frac{1}{2}}$}\pi$, where
the coefficients $c_{r}(0)$ are obtained from (6.4) with $\beta=b$. The
expansion valid in $|\arg\,z|<\mbox{${\textstyle\frac{1}{2}}$}\pi$ follows
from (6.8) and the Kummer-type transformation $J(a,b;z)=e^{z}J(b,a;-z)$.

## References

  * [1] M.A. Chaudhry, A. Qadir, H.M. Srivastava and R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004) 589–602.
  * [2] S.A. Dar and R.B. Paris, On the properties of the $(p,\nu)$-extension of the Whittaker function $M_{\kappa,\mu}(z)$, J. Ramanujan Math. Soc. (2021) (to appear). [arXiv:1801.09921].
  * [3] I.S. Gradshteyn and I.M. Rhyzik, Table of Integrals, Series and Products, Academic Press, New York, 1980.
  * [4] A.A. Kilbas, R.K. Saxena and J.J. Trujillo, Krätzel function as a function of hypergeometric type, Frac. Calc. Appl. Anal. 9 (2006) 109–131.
  * [5] E. Krätzel, Integral transformations of Bessel type,. In: Generalized Functions and Operational Calculus (Proc. Conf. Varna 1975), Bulg. Acad. Sci., Sofia (1979), 148–155.
  * [6] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974; Reprinted in A.K. Peters, Massachussets, 1997.
  * [7] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
  * [8] R.B. Paris, Asymptotics of the special functions of fractional calculus, In: Handbook of Fractional Calculus with Applications, Vol. 1, (Eds. A. Kochubei, Y. Luchko), pp. 297–323, De Gruyter, Berlin, 2019.
  * [9] R.B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2001.
  * [10] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.