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arXiv · 2609.00074

Double basic hypergeometric sums via a regularized Jackson $q$-integral

Takumi Maesaka
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Abstract

Using the Andrews--Askey integral, we derive a holomorphic extension formula for a Jackson $q$-integral. This formula allows identities established for terminating specializations to be continued to the nonterminating case. Our first application yields two companion binomial-type double-sum formulas. The second formula contains, as special cases, the nonterminating Sears--Carlitz transformation of Gasper and Rahman and Rahman's generating function for the Askey--Wilson polynomials. Applying the same method to the $q$-Chu--Vandermonde and Rogers' ${}_6\phi_5$ summations, we obtain two further integral representations for double sums of basic hypergeometric type. A balanced ${}_4\phi_3$ specialization of the latter identity is equivalent, via Sears' transformation, to the double-series transformation of Ismail, Rahman, and Suslov.

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Takumi Maesaka. 2026-08-31. Double basic hypergeometric sums via a regularized Jackson $q$-integral. https://arxiv.org/abs/2609.00074

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