Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel

Abstract

We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.