Sums of squares and products of Bessel functions (Q1789491)

Let \(r_k(n)\) denote the number of representations of the positive integer \(n\) as a sum of \(k\) squares. The authors provide a rigorous proof of a Voronoï summation formula for \(r_k(n)\), \(k\geq 2\). Using this summation formula, they establish a new transformation between a series consisting of \(r_k(n)\) and a product of two Bessel functions, and a series involving \(r_k(n)\) and the Gaussian hypergeometric function. This transformation can be considered as a broad generalization of well-known results of \textit{G. H. Hardy} [Quart. J. Pure Appl. Math. 46, 263--283 (1915; JFM 45.1253.01)], and of \textit{A. L. Dixon} and \textit{W. L. Ferrar} [Quart. J. Math., Oxford Ser. 5, 48--63 (1934; Zbl 0009.00901)], as well as of a classical result of \textit{A. I. Popov} [C.R. Acad. Sci. URSS 2, 96--99 (1935; Zbl 0011.35501)].