Gosper's and Ramanujan's infinite products and generalizations

Abstract

Gosper introduced many remarkable formulas involving infinite products. One such formula provides an evaluation of the product $$\prod_{n \geq 1} \frac{1}{e}\left(\frac{1}{3n} + 1\right)^{3n + \frac{1}{2}}. $$ It appears that a complete proof of this evaluation has not appeared in the literature. In this paper, we introduce some techniques to extend Gosper's evaluation. We prove a generalization of Gosper's formula involving three free parameters, which can also be applied to evaluate arbitrary multisections of Gosper's product. Then we investigate higher-order extensions involving higher-degree polynomials in $n$, relative to the polynomials $3n$ and $3n+\frac{1}{2}$ involved in Gosper's product, yielding a generalization of an infinite product evaluation due to Ramanujan and recently studied by Bradley--Thrush [\textit{\it Ramanujan J.} (2025)]. We also consider connections with infinite products due to Kurokawa and Rovinski\uı, building on the recent work of the first and fourth authors [\textit{\it Ramanujan J.} (2025)].