An analogue of a formula of Popov II

TL;DR

This paper generalizes a summation formula involving the number of representations of integers as sums of squares, extending Bessel functions to Whittaker functions with a novel proof approach.

Contribution

It introduces a new generalized summation formula involving $r_k(n)$ and Whittaker functions, expanding previous results with a different proof method.

Findings

Generalization of a summation formula involving $r_k(n)$ and Bessel functions.

Extension of Bessel functions to Whittaker functions in the formula.

A drastically different proof technique from previous work.

Abstract

Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a generalization of a summation formula already proved by us [Advances in Applied Mathematics, 175 (2026) 103201], which involves the arithmetical function $r_{k}(n)$ and the Bessel functions of the first kind. We extend the Bessel functions in the aforementioned formula to Whittaker functions, and our proof of this generalization is drastically different from the proof of the particular case presented in [Advances in Applied Mathematics, 175 (2026) 103201].