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

Abstract

We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling polynomials is then used to obtain recurrence relations, gamma--polygamma formulas for the polynomials $P_{m-s}(m,x)$, a more transparent proof of a vanishing identity used in earlier closed forms, and a finite approximation to $\cosh πx$ with a corresponding limit formula for $π$. We also observe that these polynomials occur naturally as signed residues of the equal-period Barnes multiple zeta function, namely $P_k(m,x)=(-1)^k m!\operatorname*{Res}_{s=m+1-k}ζ_{m+1}(s,x)$. In addition, we derive the reflection formula $P_k(m,m+1-x)=(-1)^kP_k(m,x)$ and use these polynomial identities to obtain explicit identities for Stirling cycle numbers. We then turn to finite nested sums built from the hyperbolic-secant integral sequence $χ_n$. After the lower bounds are fixed, the nested sums become coefficient-counting problems: the common-lower-bound case gives binomial coefficients, while the staircase case gives Catalan numbers. Combining these counts with the closed forms for the individual $χ_j$'s produces explicit evaluations involving Catalan's constant, zeta values, and polygamma values at one quarter. A Wolfram Language package accompanies the formulas.