Signed generalized Stirling polynomials, nested sums, and hyperbolic secant integral identities

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 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.