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

Abstract

AbstractWe 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 $\chi (n)$ .