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

Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\le n<\infty$, generated by Dirichlet series $$\sum _{n=1}^{\infty }\frac{a(n)}{{\lambda }_n^s}\text{and}\sum _{n=1}^{\infty }\frac{b(n)}{{\mu }_n^s},$$ 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 ${}_2{F}_1(a,b;c;z)$. 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 $\chi (n)$.

24 pages, submitted for publication. arXiv admin note: text overlap with arXiv:2108.13991