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 ∏n ≥ 1 1/e(1/3n + 1)3n + 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+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)].
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BibTeX
@misc{allouche2026gospers,
title = {{Gosper's and Ramanujan's infinite products and generalizations}},
author = {Jean-Paul Allouche and John M. Campbell and Simon R. Holcombe and Lubomir Markov and Manon Stipulanti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03238},
eprint = {2610.03238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03238},
note = {CC0},
}
RIS
TY - UNPB
TI - Gosper's and Ramanujan's infinite products and generalizations
AU - Jean-Paul Allouche
AU - John M. Campbell
AU - Simon R. Holcombe
AU - Lubomir Markov
AU - Manon Stipulanti
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03238
UR - https://arxiv.org/abs/2610.03238
AB - Gosper introduced many remarkable formulas involving infinite products. One such formula provides an evaluation of the product ∏_n ≥ 1 1/e(1/3n + 1)^3n + 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+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)].
ER -