Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities
Abstract
We study signed generalized Stirling polynomials Pk(m,x) arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma--polygamma formulas for Pm−s(m,x), a central vanishing identity, a finite approximation to cosh πx, and a limit formula for π. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner--Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence χn. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for χj yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.
Related
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- The Generalized Lax Conjecture for Strictly Hyperbolic Polynomials
- Evaluation of a Double Integral with Exponentials and Two Generalized Laguerre Polynomials and Its Associated Moments
- Weak Hurwitz stability of extended Turán expressions for Stirling--Whitney--Riordan and Laguerre polynomials
- Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials
BibTeX
@misc{abdulsalam2026signed,
title = {{Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities}},
author = {Abdulhafeez A. Abdulsalam and Michael J. Schlosser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.26846},
eprint = {2605.26846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26846},
}
RIS
TY - UNPB
TI - Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities
AU - Abdulhafeez A. Abdulsalam
AU - Michael J. Schlosser
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2605.26846
UR - https://arxiv.org/abs/2605.26846
AB - We study signed generalized Stirling polynomials P_k(m,x) arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma--polygamma formulas for P_m−s(m,x), a central vanishing identity, a finite approximation to cosh πx, and a limit formula for π. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner--Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence χ_n. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for χ_j yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.
ER -