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Two general series identities involving modified Bessel functions and a class of arithmetical functions

Part of: Zeta and $L$-functions: analytic theory Multiplicative number theory

Published online by Cambridge University Press:  10 October 2022

Bruce C. Berndt ,
Atul Dixit ,
Rajat Gupta  and
Alexandru Zaharescu
Bruce C. Berndt
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA e-mail: berndt@illinois.edu
Atul Dixit
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat 382355, India e-mail: adixit@iitgn.ac.in
Rajat Gupta*
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat 382355, India. Current address: Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 106319, Taiwan
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA, and Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest RO-70700, Romania e-mail: zaharesc@illinois.edu
*
e-mail: rajatgpt@gate.sinica.edu.tw
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Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty $, generated by Dirichlet series

$$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$

satisfying a familiar functional equation involving the gamma function $\Gamma (s)$. Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$, and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$, the Bessel functions of imaginary argument $I_{\mu }(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$, the number of representations of n as a sum of k squares $r_k(n)$, and primitive Dirichlet characters $\chi (n)$.

Keywords

Bessel functionsfunctional equationsclassical arithmetic functions

MSC classification

Primary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11N99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 6 , December 2023 , pp. 1800 - 1830
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The first and second authors sincerely thank the MHRD SPARC project SPARC/2018-2019/P567/SL for their financial support. The first author is also supported by a grant from the Simons Foundation. The third author is a postdoctoral fellow at IIT Gandhinagar supported, in part, by the grant CRG/2020/002367.

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