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

Abstract

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 $\cosh \pi x$, and a limit formula for $\pi$. 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 ${\chi }_n$. 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 ${\chi }_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.

34 pages; added consequences of Yakubovich's integral evaluations