Applications of Tannery's theorem to Bailey's ψ66 summation

Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leadin...

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Bibliographic Details
Published in:	Journal of mathematical analysis and applications Vol. 541; no. 1
Main Author:	Bradley-Thrush, J.G.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.01.2025
Subjects:	Basic hypergeometric series Special functions Tannery's theorem Basic hypergeometric series Tannery's theorem Special functions
ISSN:	0022-247X
Online Access:	Get full text
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Summary:	Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leading to the identification of two counterintuitive instances of series for which interchanging the limit and the summation yields incorrect results. Analogous limiting processes involving ordinary hypergeometric series are considered.
ISSN:	0022-247X
DOI:	10.1016/j.jmaa.2024.128659