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, including a comparison with centered Meixner-Pollaczek polynomials. We also 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.
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