An analogue of a formula of Popov II

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].