Sums of squares and products of Bessel functions

Bruce C. Berndt, Atul Dixit, Sun Kim, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

Let rk(n) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoï summation formula for rk(n),k≥2, proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of rk(n) and a product of two Bessel functions, and a series involving rk(n) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.

Original languageEnglish (US)
Pages (from-to)305-338
Number of pages34
JournalAdvances in Mathematics
Volume338
DOIs
StatePublished - Nov 7 2018

Keywords

  • Analytic continuation
  • Bessel functions
  • Sums of squares
  • Voronoï summation formula

ASJC Scopus subject areas

  • Mathematics(all)

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