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 $\operatorname{sech}x$. 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.