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Choose which manuscript you would like to read first: the signed Stirling polynomials project or the well-poised q-Taylor expansions project.

01
Pk(m,x)

Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities

We begin with signed generalized Stirling polynomials arising from a generalization of Malmsten's integral and show how their structure gives recurrence relations, gamma-polygamma formulas, a finite approximation to \(\cosh \pi x\), signed residues of the Barnes multiple zeta function, reflection identities, and explicit Stirling cycle number identities. We then turn to finite nested sums built from the hyperbolic-secant integral sequence \(\chi_n\), where fixed lower bounds lead to binomial and Catalan-number coefficient counts.

Malmsten integral hyperbolic secant integrals signed generalized Stirling polynomials symmetric functions Barnes multiple zeta function Hurwitz zeta function nested sums
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02
q

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

We prove a nonterminating well-poised basic \(q\)-Taylor expansion with complementary remainders for a two-basis infinite-product kernel. The well-poised parameter gives the rational basic basis, while the elliptic nome is treated as a separate deformation. 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 \({}_8\phi_7\) summation, Rogers' \({}_6\phi_5\) summation, and theta interpolation, with Bailey's nonterminating \({}_8\phi_7\) summation recovered as a consequence.

Askey-Wilson operator q-Taylor expansion well-poised basic hypergeometric series two-basis kernel remainder term nonterminating \({}_8\phi_7\) summation
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