Double basic hypergeometric sums via a regularized Jackson q-integral
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' 6ϕ5 summations, we obtain two further integral representations for double sums of basic hypergeometric type. A balanced 4ϕ3 specialization of the latter identity is equivalent, via Sears' transformation, to the double-series transformation of Ismail, Rahman, and Suslov.
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BibTeX
@misc{maesaka2026double,
title = {{Double basic hypergeometric sums via a regularized Jackson $q$-integral}},
author = {Takumi Maesaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00074},
eprint = {2609.00074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00074},
}
RIS
TY - UNPB
TI - Double basic hypergeometric sums via a regularized Jackson q-integral
AU - Takumi Maesaka
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00074
UR - https://arxiv.org/abs/2609.00074
AB - 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' _6ϕ_5 summations, we obtain two further integral representations for double sums of basic hypergeometric type. A balanced _4ϕ_3 specialization of the latter identity is equivalent, via Sears' transformation, to the double-series transformation of Ismail, Rahman, and Suslov.
ER -