Transformations involving \(r_k(n)\) and Bessel functions
January 10, 2017, 3:00 — 3:50 pm at LIT 368.
Let \(r_k(n)\) denote the number of representations of the positive integer \(n\) as the sum of \(k\) squares, where \(k\geq 2\). In 1934, the Russian mathematician Alexander Ivanovich Popov, who is more popularly known as one of the world experts in Finno-Ugric Linguistics, obtained a beautiful transformation between two series involving \(r_k(n)\) and Bessel functions. Unfortunately, Popov’s proof appears to be defective since there are subtleties involved in extending the available results on \(r_2(n)\) to those involving \(r_k(n)\), \(k>2\), and since the usual techniques do not carry over. In this work, we give a rigorous proof of Popov’s result by observing that N. S. Koshliakov’s ingenious proof of the Voronoi summation formula for coefficients of a Dirichlet series satisfying a functional equation with one gamma factor circumvents these difficulties. We then obtain an analogue of a double Bessel series identity on page 335 of Ramanujan’s Lost Notebook in the spirit of Popov’s identity.
In the second part of our talk, we will obtain a proof of a more general summation formula for \(r_k(n)\) due to A. P. Guinand, which is claimed in his work without proof and without any conditions, and apply it to obtain a new transformation of a series involving \(r_k(n)\) and a product of two Bessel functions. This transformation can be considered as a massive generalization of many well-known results in the literature, for example, those of A.L. Dixon and W.L. Ferrar, of G.H. Hardy, and of a classical result of Popov. This is joint work with Bruce C. Berndt, Sun Kim and Alexandru Zaharescu.