On a double series
Aung Phone Maw
TL;DR
This paper investigates a two-parameter double series with a Bose–Einstein denominator, defined by $f(x,y;w)=\sum_{n,r\ge1} \frac{1}{\sqrt{x^2 n^2 + r^2 + w^2} ( e^{2\pi y \sqrt{x^2 n^2 + r^2 + w^2}} - 1 )}$. It derives a broad functional relation for $f$ under scalable transformations of $(x,y,w)$ (Proposition 1), using a chain of substitutions and summations, and then specializes to $w=0$ to obtain a symmetric identity. A corollary expresses sums involving representations counted by $r_{a,b}(n)$ in terms of parameter ratios, illustrating a lattice-point counting connection. The results provide a modular-like transform for lattice-sum-type series and contribute tools for evaluating and relating Bose–Einstein–type double sums, with potential applications in analytic number theory and spectral sums.
Abstract
We shall investigate and arrive at a certain functional property of the double series \[ \sum\limits_{n,r\geq 1}\frac{1}{\sqrt{x^2n^2+r^2+w^2}\left( e^{2 πy\sqrt{x^2n^2+r^2+w^2}}-1\right)}. \]