# Atul Dixit

## Transformations involving $$r_k(n)$$ and Bessel functions

#### When/Where:

January 10, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

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.