Higher-order tangent and secant numbers, special-value integrals, and a common zero attractor
Abstract
We study two families of coefficients associated with powers of the tangent function and with products of the secant and tangent functions, and develop a unified treatment of the corresponding higher-order tangent and secant numbers. The first part establishes a common identity for even derivatives of tanh x and odd derivatives of sechx. Combined with integral formulae for these derivatives, this yields finite power-integral representations for the normalized values ζ(2n+1) and β(2n) and naturally introduces two polynomial families Pn and Qn. We determine several arithmetic and structural properties of these polynomials. In particular, we identify the exact content of Pn, prove that Qn is primitive, establish strict log-concavity and unimodality of their coefficient sequences, and describe their real zeros. For even indices each family has a unique negative real zero, and the two sequences of real zeros have the same limit. We also derive differential recurrences and generating functions for Pn and Qn. The generating functions lead to a global complex-singularity analysis. We construct a compact Jordan curve Γ and prove that the complex zero sets of both Pn and Qn converge to Γ in Hausdorff distance. The normalized zero-counting measures converge weakly to the same equilibrium measure on Γ. We determine the logarithmic capacity of the enclosed compact set, give an explicit arclength density for the limiting measure, and compute its first moments. Numerical plots for P20, P50, P100 and P150 illustrate the zero-attractor theorem.
Related
- Uniqueness of tangent cones with immersed special Legendrian links
- A Proof of the Global Attractor Conjecture in a Special Case
- Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals
- Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals
- Higher-order tangent lines to smooth hypersurfaces
BibTeX
@misc{waffo2026higherorder,
title = {{Higher-order tangent and secant numbers, special-value integrals, and a common zero attractor}},
author = {Luc Ramsès Talla Waffo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30208},
eprint = {2610.30208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30208},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-order tangent and secant numbers, special-value integrals, and a common zero attractor
AU - Luc Ramsès Talla Waffo
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30208
UR - https://arxiv.org/abs/2610.30208
AB - We study two families of coefficients associated with powers of the tangent function and with products of the secant and tangent functions, and develop a unified treatment of the corresponding higher-order tangent and secant numbers. The first part establishes a common identity for even derivatives of tanh x and odd derivatives of sechx. Combined with integral formulae for these derivatives, this yields finite power-integral representations for the normalized values ζ(2n+1) and β(2n) and naturally introduces two polynomial families P_n and Q_n. We determine several arithmetic and structural properties of these polynomials. In particular, we identify the exact content of P_n, prove that Q_n is primitive, establish strict log-concavity and unimodality of their coefficient sequences, and describe their real zeros. For even indices each family has a unique negative real zero, and the two sequences of real zeros have the same limit. We also derive differential recurrences and generating functions for P_n and Q_n. The generating functions lead to a global complex-singularity analysis. We construct a compact Jordan curve Γ and prove that the complex zero sets of both P_n and Q_n converge to Γ in Hausdorff distance. The normalized zero-counting measures converge weakly to the same equilibrium measure on Γ. We determine the logarithmic capacity of the enclosed compact set, give an explicit arclength density for the limiting measure, and compute its first moments. Numerical plots for P_20, P_50, P_100 and P_150 illustrate the zero-attractor theorem.
ER -