Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions

TL;DR

This paper establishes two new general identities involving modified Bessel functions, hypergeometric functions, and arithmetical functions, derived from Dirichlet series satisfying a functional equation with gamma functions.

Contribution

It introduces novel identities linking Bessel functions and hypergeometric functions with arithmetical functions, expanding the theoretical framework of modular relations.

Findings

Derived two general identities involving Bessel and hypergeometric functions.

Identified connections between arithmetical functions and special functions.

Extended known special cases to more general identities.

Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ 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 $\chi(n)$.