An expansion for the sum of a product of an exponential and a Bessel function

TL;DR

This paper derives convergent asymptotic expansions for sums involving exponential and Bessel functions, useful for evaluating these sums as parameters approach specific limits, with extensions to modified Bessel functions and alternating sums.

Contribution

It introduces a novel double Mellin-Barnes integral method to obtain convergent asymptotic expansions for sums of exponential and Bessel functions, including modified and alternating cases.

Findings

Derived convergent asymptotic expansion for the sum with Bessel functions.

Extended the method to sums with modified Bessel functions.

Provided analysis for the sum's behavior as parameter a approaches zero.

Abstract

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind of order $\nu>-1/2$ and $a$, $b$ are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$ with $b<2\pi$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_\nu(x)$. The alternating versions of these sums are also mentioned.

Equations

$$\sum _{n=1}^{\infty }\frac{e^{-an}}{(\text{leavevmode}\frac{1}{2}bn{)}^{\nu }}{J}_{\nu }(bn),$$

$$S_{J,\nu }(a,b)=\sum _{n=1}^{\infty }\frac{e^{-an}}{(\text{leavevmode}\frac{1}{2}bn{)}^{\nu }}{J}_{\nu }(bn),\text{leavevmode}and{S}_{K,\nu }(a,b)=\sum _{n=1}^{\infty }\frac{e^{-an}}{(\text{leavevmode}\frac{1}{2}bn{)}^{\nu }}{K}_{\nu }(bn),$$

$$S_{J,\nu }(a,b)=\sum _{n=1}^{\infty }\frac{e^{-an}}{(\text{leavevmode}\frac{1}{2}bn{)}^{\nu }}{J}_{\nu }(bn)(a>0,b>0),$$

$$S_{J,\nu }(0,b)=\frac{\sqrt{\pi }}{b\Gamma (\nu +\text{leavevmode}\frac{1}{2})}-\frac{1}{2\Gamma (1+\nu )}(\nu >-\text{leavevmode}\frac{1}{2},0<b<2\pi ).$$

$$S_{J,\nu }(a,b)=\sum _{n=1}^{\infty }{e}^{-an}\sum _{k=0}^{\infty }\frac{(-{)}^k(\text{leavevmode}\frac{1}{2}bn{)}^{2k}}{k!\Gamma (1+\nu +k)}=\sum _{k=0}^{\infty }\frac{(-{)}^k(\text{leavevmode}\frac{1}{2}b{)}^{2k}}{k!\Gamma (1+\nu +k)}\sum _{n=1}^{\infty }{e}^{-an}{n}^{2k}$$

$$S_{J,\nu }(a,b)=\sum _{k=0}^{\infty }\frac{(-{)}^k(\text{leavevmode}\frac{1}{2}b{)}^{2k}}{k!\Gamma (1+\nu +k)}\text{leavevmode}{Li}_{-2k}(e^{-a}).$$

$$\text{leavevmode}{Li}_{-n}(e^{-a})$$

$$\text{leavevmode}{Li}_{-n}(e^{-a})\sim n!a^{-n-1}(n\to \infty ).$$

$$\frac{1}{a}\,\frac{(-)^{k}\Gamma(2k+1)}{k!\Gamma(1+\nu+k)}\,\biggl{(}\frac{b}{2a}\biggr{)}^{\!2k}=\frac{(-)^{k}\Gamma(k+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}{a\sqrt{\pi}\,\Gamma(1+\nu+k)}\,\biggl{(}\frac{b}{a}\biggr{)}^{\!2k}=(-)^{k}O\biggl{(}k^{-\nu-1/2}\biggl{(}\frac{b}{a}\biggr{)}^{\!2k}\biggr{)}$$

$$e^{-x}=\frac{1}{2\pi i}{\int }_{c-\infty i}^{c+\infty i}\Gamma (s)x^{-s}ds(|\arg x|<\text{leavevmode}\frac{1}{2}\pi )$$

$$(\text{leavevmode}\frac{1}{2}x{)}^{-\nu }{J}_{\nu }(x)=\frac{1}{2\pi i}{\int }_{c-\infty i}^{c+\infty i}\frac{\Gamma (s)}{\Gamma (1+\nu -s)}(\text{leavevmode}\frac{1}{2}x{)}^{-2s}ds(x>0,\nu >0),$$

$$S_{J,\nu }(a,b)$$

$$T_1$$

$$\zeta (s)=2^s{\pi }^{s-1}\zeta (1-s)\Gamma (1-s)\sin \text{leavevmode}\frac{1}{2}\pi s,$$

$$T_{1}=-\frac{1}{2\Gamma(1\!+\!\nu)}+\frac{a}{2\pi^{2}}\sum_{m,n\geq 0}\frac{(-)^{n}\cos\pi(m\!+\!n)}{(2m\!+\!1)!n!}\,\frac{\Gamma(2m\!+\!2n\!+\!2)}{\Gamma(n\!+\!1\!+\!\nu)}\,\zeta(2m\!+\!2n\!+\!2)\,\biggl{(}\frac{a}{2\pi}\biggr{)}^{\!2m}\biggl{(}\frac{b}{2\pi}\biggr{)}^{\!2n}$$

$$=-\frac{1}{2\Gamma(1\!+\!\nu)}+\frac{1}{\pi}\sum_{m,n\geq 0}(-)^{m}A_{m,n}\,\biggl{(}\frac{a}{2\pi}\biggr{)}^{\!2m+1}\biggl{(}\frac{b}{2\pi}\biggr{)}^{\!2n},$$

$$A_{m,n}:=\frac{\Gamma (m+n+1)\Gamma (m+n+\text{leavevmode}\frac{3}{2})}{m!n!\Gamma (m+\text{leavevmode}\frac{3}{2})\Gamma (n+1+\nu )}\zeta (2m+2n+2).$$

$$\frac{a^{-1}}{2\pi i}\int_{-\infty i}^{\infty i}\frac{\Gamma(s)\Gamma(1-2s)}{\Gamma(1+\nu+s)}\,\biggl{(}\frac{b}{2a}\biggr{)}^{\!2s}ds$$

$$T_{2}=\frac{1}{b}\sum_{k=0}^{\infty}\frac{(-)^{k}\Gamma(\leavevmode\hbox{${\textstyle\frac{1}{2}}$}k+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}{k!\Gamma(\leavevmode\hbox{${\textstyle\frac{1}{2}}$}+\nu-\leavevmode\hbox{${\textstyle\frac{1}{2}}$}k)}\,\biggl{(}\frac{2a}{b}\biggr{)}^{\!k}.$$

$$\frac{(-)^{k}\Gamma(\leavevmode\hbox{${\textstyle\frac{1}{2}}$}k+\leavevmode\hbox{${\textstyle\frac{1}{2}}$}-\nu)}{\sqrt{\pi}\,\Gamma(\leavevmode\hbox{${\textstyle\frac{1}{2}}$}k+1)}\,\biggl{(}\frac{a}{b}\biggr{)}^{\!k}\cos\pi(\nu-\leavevmode\hbox{${\textstyle\frac{1}{2}}$}k)=(-)^{k}O\biggl{(}k^{-\nu-1/2}\biggl{(}\frac{a}{b}\biggr{)}^{\!k}\biggr{)}\qquad(k\to\infty),$$

$$T_{2}=\frac{\sqrt{\pi}}{b\Gamma(\nu+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}\biggl{(}1+\frac{a^{2}}{b^{2}}\biggr{)}^{\!\nu-\frac{1}{2}}\!-\frac{2a}{b^{2}\Gamma(\nu)}\,{}_{2}F_{1}\biggl{(}1,1-\nu;\frac{3}{2};-\frac{a^{2}}{b^{2}}\biggr{)}.$$

$$T_{2}=\frac{\sqrt{\pi}}{b\Gamma(\nu+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}\biggl{(}1+\frac{a^{2}}{b^{2}}\biggr{)}^{\!\nu-\frac{1}{2}}\!-\frac{2a}{(a^{2}+b^{2})\Gamma(\nu)}\,{}_{2}F_{1}\biggl{(}1,\leavevmode\hbox{${\textstyle\frac{1}{2}}$}+\nu;\frac{3}{2};\frac{a^{2}}{a^{2}+b^{2}}\biggr{)}.$$

$$S_{J,\nu}(a,b)=\frac{\sqrt{\pi}}{b\Gamma(\nu+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}\biggl{(}1+\frac{a^{2}}{b^{2}}\biggr{)}^{\!\nu-\frac{1}{2}}\!-\frac{1}{2\Gamma(1+\nu)}-\frac{2a}{b^{2}\Gamma(\nu)}\,{}_{2}F_{1}\biggl{(}1,1-\nu;\frac{3}{2};-\frac{a^{2}}{b^{2}}\biggr{)}$$

$$+\frac{a}{2\pi^{2}}\sum_{m,n\geq 0}(-)^{m}A_{m,n}\,\biggl{(}\frac{a}{2\pi}\biggr{)}^{\!2m}\biggl{(}\frac{b}{2\pi}\biggr{)}^{\!2n},$$

$$T_{1}=-\frac{1}{2\Gamma(1+\nu)}+\frac{1}{\pi}\sum_{k=0}^{\infty}(-)^{k}C_{k}\biggl{(}\frac{a}{2\pi}\biggr{)}^{\!2k+1},$$

$$C_{k}=\sum_{n=0}^{\infty}A_{k,n}\biggl{(}\frac{b}{2\pi}\biggr{)}^{\!2n}\qquad(b<2\pi).$$

$$S_{J,\nu}(a,b)=\frac{\sqrt{\pi}}{b\Gamma(\nu+\leavevmode\hbox{${\textstyle\frac{1}{2}}$})}\biggl{(}1+\frac{a^{2}}{b^{2}}\biggr{)}^{\!\nu-\frac{1}{2}}\!-\frac{1}{2\Gamma(1+\nu)}-\frac{2a}{b^{2}\Gamma(\nu)}\,{}_{2}F_{1}\biggl{(}1,1-\nu;\frac{3}{2};-\frac{a^{2}}{b^{2}}\biggr{)}$$

$$+\frac{1}{\pi}\sum_{k=0}^{\infty}(-)^{k}C_{k}\,\biggl{(}\frac{a}{2\pi}\biggr{)}^{\!2k+1},$$

$${\hat{S}}_{J,\nu }(a,b)=\sum _{n=1}^{\infty }\frac{(-{)}^{n-1}{e}^{-an}}{(\text{leavevmode}\frac{1}{2}bn{)}^{\nu }}{J}_{\nu }(bn).$$

$${\hat{S}}_{J,\nu }(a,b)=S_{J,\nu }(a,b)-S_{J,\nu }(2a,2b),$$

Full text

[TABLE]

where $J_{\nu}(x)$ is the Bessel function of the first kind of order \nu>-\leavevmode\hbox{{\textstyle\frac{1}{2}}} and $a$, $b$ are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$ with $b<2\pi$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_{\nu}(x)$. The alternating versions of these sums are also mentioned.

Mathematics Subject Classification: 33C05, 33C10, 33C20, 41A30, 41A60

Keywords: Bessel functions, Mellin-Barnes integral, asymptotic expansion

1. Introduction

We consider the sums

[TABLE]

where $J_{\nu}(x)$ is the Bessel function of the first kind and $K_{\nu}(x)$ the modified Bessel function. The parameters $a$, $b$ are assumed to be positive and the order \nu>-\leavevmode\hbox{{\textstyle\frac{1}{2}}} for $S_{J,\nu}(a,b)$ and $\nu\geq 0$ for $S_{K,\nu}(a,b)$. Convergent expansions are derived for these sums that involve the polylogarithm function $\leavevmode\hbox{Li}_{s}(x)$ of negative integer order $s$ when $a>b$ and the Riemann zeta function when $a\leq b$; the latter case requires $a,b\in(0,2\pi)$ to secure convergence. The evaluation of the alternating versions of (1.1) follows in a straightforward manner in terms of these sums with parameters $a$, $b$ and $2a$, $2b$.

Our main interest in the sums in (1.1) is the limit as $a\to 0$ when convergence of the first sum becomes slow. When $a=0$, the sum $S_{J,\nu}(0,b)$ has been considered by Tric̆ković et al. in [4], where approaches using Poisson’s summation formula and Bessel’s integral were employed to derive convergent expansions. The sums $S_{J,\nu}(0,b)$ and $S_{K,\nu}(0,b)$ have been discussed in [2] using a Mellin transform approach. With the presence of the exponential factor in the above sums it is found that a double Mellin-Barnes integral is required to derive an expansion as $a\to 0$. The treatment of such integrals has been discussed in [3, Chapter 7] in the context of multi-dimensional Laplace-type integrals.

2.  A series representation for $S_{J,\nu}(a,b)$ valid when $a\geq b$

We consider the sum

[TABLE]

where $J_{\nu}(x)$ denotes the usual Bessel function of real order \nu>-\leavevmode\hbox{{\textstyle\frac{1}{2}}}. In the case $a=0$, the sum has the value [2, Theorem 1, (2.8)]

[TABLE]

Expanding the Bessel function as a series, we have

[TABLE]

upon interchanging the order of summation. If we introduce the polylogarithm function $\leavevmode\hbox{Li}_{s}(x)$ defined by $\leavevmode\hbox{Li}_{s}(z)=\sum_{n=1}^{\infty}n^{-s}z^{n}$ [1, (25.12.10)], we then obtain

[TABLE]

The polylogarithm function of negative integer order $-n$, $n=0,1,2,\ldots$ is

[TABLE]

with $\phi_{k}:=\arctan(2\pi k/a)$. For large $n$ with $a$ fixed we have

[TABLE]

The late terms in the sum (2.3) consequently possess the behaviour

[TABLE]

as $k\to\infty$, so that the sum (2.3) converges when $a>b$, and when $a=b$ provided \nu>-\leavevmode\hbox{{\textstyle\frac{1}{2}}}. Calculations with Mathematica confirm the validity of this statement, where it should be noted that $\leavevmode\hbox{Li}_{s}(z)$ is a built-in library function.

3.  A series representation of $S_{J,\nu}(a,b)$ for $a\to 0$

To deal with the case $a<b$, and in particular to obtain an expansion for $S_{J,\nu}(a,b)$ as $a\to 0$, we employ the Cahen-Mellin integral [3, p. 89]

[TABLE]

and the Mellin-Barnes integral [1, (10.9.22)]

[TABLE]

where $c>0$ in both cases so that the integration path lies to the right of the poles of the integrand situated at s=0,-1,-2,\ldots\. Combination of these two results leads to the double Mellin-Barnes integral representation for $S_{J,\nu}(a,b)$ given by, provided222If the integation path in (3.2) is bent back into a loop that encloses the poles of $\Gamma(s)$ with endpoints at infinity in $\Re(s)<0$, then the integral holds without restriction on $\nu$; see [3, p. 115]. $\nu>0$,

[TABLE]

where $\zeta(s)$ is the Riemann zeta function. The integration contours are indented to pass to the right of the pole of the zeta function and those of $\Gamma(s)$ and $\Gamma(t)$.

Let us first consider the poles at $s=-n$, $t=-m$ ($m,n=0,1,2,\ldots$) that result from the gamma functions in the numerator of (3.3). Displacement of the integration paths to the left produces the formal series

[TABLE]

where even values of $m$ (apart from $m=0$ when $n=0$) do not contribute to the double sum on account of the trivial zeros of $\zeta(s)$ at s=-2,-4,\ldots\. Using the functional relation for $\zeta(s)$ [1, (25.4.2)]

[TABLE]

the value \zeta(0)=-\leavevmode\hbox{{\textstyle\frac{1}{2}}} and the duplication formula \Gamma(2z)=2^{2z-1}\pi^{-1/2}\Gamma(z)\Gamma(z+\leavevmode\hbox{{\textstyle\frac{1}{2}}}), we find that

[TABLE]

[TABLE]

where

[TABLE]

Application of Stirling’s formula for the gamma function, together with the fact that $\zeta(2m+2n+2)=O(1)$ for large $m$, $n$, shows that $A_{m,n}$ has only algebraic growth in $m$ and $n$ as $m,n\to\infty$. Consequently the double sum in (3.4) converges when $a<2\pi$ and $b<2\pi$.

We now deal with the pole of the zeta function when $2s+t=1$ with unit residue. Setting $t=1-2s$, we find the contribution

[TABLE]

where the integration path is indented to separate the poles of $\Gamma(s)$ and $\Gamma(1-2s)$. Evaluation of the residues of $\Gamma(1-2s)$ at s=\leavevmode\hbox{{\textstyle\frac{1}{2}}}+\leavevmode\hbox{{\textstyle\frac{1}{2}}}k, $k=0,1,2,\ldots$ leads to the contribution

[TABLE]

The terms in this sum behave like

[TABLE]

so that (3.6) converges for $a<b$, and for $a=b$ provided \nu>-\leavevmode\hbox{{\textstyle\frac{1}{2}}}.

By separating terms with even and odd $k$, we can express $T_{2}$ in terms of well-known special functions to yield

[TABLE]

Alternatively, the hypergeometric function can be written in a different form using Euler’s transformation [1, (15.8.1)] to yield

[TABLE]

To summarise, we have the following expansion.

[TABLE]

[TABLE]

provided $a<2\pi$ and $b<2\pi$. The coefficients $A_{m,n}$ are defined in (3.5), .

When $a=0$ it is seen that (3.8) correctly reduces to (2.2).

The above expansion can be arranged in a form suitable for asymptotic calculations in the limit $a\to 0$. We note that the series representation of the hypergeometric function in (3.7) is already in the form of a convergent asymptotic expansion as $a\to 0$. We find after routine manipulation that

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where

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Consequently we obtain

[TABLE]

[TABLE]

where the coefficients $C_{k}$ are defined in (3.9).

Finally, the alternating version of (2.1) is given by

[TABLE]

It is easily seen that

[TABLE]

from which the expansion as $a\to 0$ can be obtained from Theorem 2.

4.  The expansion involving the modified Bessel function

The same procedure can be brought to bear on the sum involving the modified Bessel function $K_{\nu}(z)$ given by

[TABLE]

where we consider only $\nu\geq 0$, since $K_{-\nu}(x)=K_{\nu}(x)$. We remove from consideration the case when $\nu$ has half-integer values when the above sum reduces to a sum of exponentials; see the appendix.

From the standard definition $K_{\nu}(x)=\pi(I_{-\nu}(x)-I_{\nu}(x))/(2\sin\pi\nu)$ valid for non-integer $\nu$ and the series expansions of $I_{\pm\nu}(x)$, we obtain

[TABLE]

when $a>b$.

Use of the integral representation333There is an error in the sector of validity in [1, (10.32.13)] and also in [3, p. 114]. [1, (10.32.13)]

[TABLE]

where $c>\max\{0,\nu\}$, together with (3.1), shows that $S_{K,\nu}(a,b)$ can be expressed as the double Mellin-Barnes integral

[TABLE]

where, as in (3.3), the integration contours are indented to pass to the right of the pole of $\zeta(2s+t)$ and those of the gamma functions. Evaluation of the residues at $s=-n$, $t=-m$ and $s=-n+\nu$, $t=-m$ ($m,n=0,1,2,\ldots$) yields, provided \nu\neq 0,\leavevmode\hbox{{\textstyle\frac{1}{2}}},1,\leavevmode\hbox{{\textstyle\frac{3}{2}}},2,\ldots\,,

[TABLE]

for $a<2\pi$ and $b<2\pi$. The coefficients $A_{m,n}$ are defined in (3.5) and

[TABLE]

To deal with the pole of the zeta function when $2s+t=1$, we obtain from (4.3)

[TABLE]

where the integration path is indented to separate the poles of $\Gamma(s)$ and $\Gamma(s-\nu)$ on the left from those of $\Gamma(1-2s)$ on the right (\nu\neq\leavevmode\hbox{{\textstyle\frac{1}{2}}},1,\leavevmode\hbox{{\textstyle\frac{3}{2}}},2,\ldots\). Evaluation of the residues on the right yields the contribution

[TABLE]

The late terms in this expansion possess the behaviour (-)^{k}(\leavevmode\hbox{{\textstyle\frac{1}{2}}}k)^{-\nu-1/2}(a/b)^{k}, so that the series in (An expansion for the sum of a product of an exponential and a Bessel function) converges for $a\leq b$ ($\nu\geq 0$).

Then we have the following expansion.

[TABLE]

[TABLE]

provided $a<2\pi$ and $b<2\pi$. The coefficients $A_{m,n}$ and $B_{m,n}$ are defined in (3.5) and (4.5).

The double sum appearing in (4.7) can be written in the alternative (asymptotic) form

[TABLE]

where the coefficients $D_{k}$ are defined by

[TABLE]

with $C_{k}$ defined in (3.9) and

[TABLE]

In the case of integer values of $\nu$ we need to take the limiting value of the double sum $T_{1}$ in (An expansion for the sum of a product of an exponential and a Bessel function). For example, if $\nu=0$ we have upon setting $\nu=\epsilon$

[TABLE]

[TABLE]

Upon noting that the hypergeometric function in (4.7) reduces to an arcsin when $\nu=0$, we then find

[TABLE]

[TABLE]

provided $a<2\pi$ and $b<2\pi$.

Finally, the alternating version of (4.1) is given by

[TABLE]

It is easily seen that

[TABLE]

from which the expansion as $a\to 0$ can be obtained from Theorem 3.

Appendix: The case of $S_{K,\nu}(a,b)$ when $\nu$ has half-integer values

When \nu=m+\leavevmode\hbox{{\textstyle\frac{1}{2}}} we have

[TABLE]

where the $c_{k}(m)$ are known constants with $c_{0}(m)=1$. Then $S_{K,\nu}(a,b)$ reduces to

[TABLE]

The sum $\sum_{n\geq 1}n^{-w}e^{-\alpha n}$ for integer $w\geq 1$ has been discussed in [3, (4.2.10)], where it is shown that

[TABLE]

where $\gamma$ is the Euler-Mascheroni constant, $\psi(x)$ is the logarithmic derivative of the gamma function and the prime on the summation sign denotes the omission of the term corresponding to $k=w-1$. From this last result it is then possible to construct the expansion of $S_{K,\nu}(a,b)$ when \nu=m+\leavevmode\hbox{{\textstyle\frac{1}{2}}}. The simplest case when \nu=\leavevmode\hbox{{\textstyle\frac{1}{2}}} yields

[TABLE]

when $a+b<2\pi$.