Abstract:We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainder for a two-basis infinite-product kernel. The finite theory is the well-poised $p=0$ specialization of the elliptic Askey--Wilson Taylor framework: the parameter $c$ produces the rational well-poised basis, while the nome $p$ gives the elliptic deformation. In the full elliptic setting with nome $p$, the Taylor expansions considered are usually finite for convergence reasons; the nonterminating expansions studied here are a distinctive feature of the basic well-poised case and do not extend directly to the elliptic setting. For the kernel proposed in the second author's earlier outlook \cite[Sec.~5]{Schlosser2008}, we compute the two Taylor coefficient families and prove the complementary limiting remainder by a Taylor-theoretic annular argument. The proof avoids Bailey's nonterminating ${}_8\phi_7$ summation; it uses instead the well-poised Cooper formula, Jackson's terminating ${}_8\phi_7$ summation, Rogers' nonterminating ${}_6\phi_5$ summation, and theta-function addition and interpolation. As a consequence, Bailey's nonterminating ${}_8\phi_7$ summation is recovered from the two-basis Taylor expansion. We also include a one-family quadratic example and a multi-kernel outlook governed by Weierstrass' elliptic partial-fraction identity.
| Comments: | 31 pages |
| Subjects: | Classical Analysis and ODEs (math.CA) |
| MSC classes: | Primary 33D15, Secondary 33D45, 39A13, 41A58 |
| Cite as: | arXiv:2605.26011 [math.CA] |
| (or arXiv:2605.26011v1 [math.CA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.26011 arXiv-issued DOI via DataCite (pending registration) |
From: Michael Schlosser [view email]
[v1]
Mon, 25 May 2026 16:29:46 UTC (32 KB)
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