Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities
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2026
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| author | Abdulsalam, Abdulhafeez A. Schlosser, Michael J. |
| author_facet | Abdulsalam, Abdulhafeez A. Schlosser, Michael J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_26846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities Abdulsalam, Abdulhafeez A. Schlosser, Michael J. Combinatorics 33B15, 05A10, 11B73, 11M35, 33E20 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. |
| title | Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities |
| topic | Combinatorics 33B15, 05A10, 11B73, 11M35, 33E20 |
| url | https://arxiv.org/abs/2605.26846 |