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

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

Research output: Contribution to journalArticlepeer-review

Abstract

We consider two sequences a(n) and b(n), 1 ≤ n < ∞, generated by Dirichlet series Σ∞ a(n)/λsn and Σ∞ b(n)μns, n=1 n=1 satisfying a familiar functional equation involving the gamma function Γ(s). Two general identities are established. The first involves the modified Bessel function Kμ(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 Kμ(z), the Bessel functions of imaginary argument Iμ(z), and ordinary hypergeometric functions 2F1(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 τ(n); the number of representations of n as a sum of k squares rk(n); and primitive Dirichlet characters x(n).

Original languageEnglish (US)
JournalCanadian Journal of Mathematics
DOIs
StateAccepted/In press - 2022

Keywords

  • Bessel functions
  • classical arithmetic functions
  • functional equations

ASJC Scopus subject areas

  • Mathematics(all)

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