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

Berndt, Bruce C. ; Dixit, Atul ; Gupta, Rajat ; Zaharescu, Alexandru

Canadian journal of mathematics, Tome 75 (2023) no. 6, pp. 1800-1830

Résumé

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 $\chi (n)$.

Détail

DOI : 10.4153/S0008414X22000530

Mots-clés : Bessel functions, functional equations, classical arithmetic functions