Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 051, 20 pages           https://doi.org/10.3842/SIGMA.2025.051
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Factorization of Basic Hypergeometric Series

Jonathan G. Bradley-Thrush
Grupo de Física Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received November 30, 2024, in final form June 22, 2025; Published online July 06, 2025

Abstract
The general problem of the factorization of a basic hypergeometric series is presented and discussed. The case of the general $_2\psi_2$ series is examined in detail. Connections are found with the theory of basic hypergeometric series on root systems. Alternative proofs of several well-known summation and transformation formulae, including Gustafson's generalization of Ramanujan's $_1\psi_1$ summation, are obtained incidentally.

Key words: basic hypergeometric series; theta functions; elliptic functions; $q$-difference equations.

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