Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals

Abstract

We develop a Dirichlet-beta counterpart and a structural extension of the coefficient-extraction framework for hyperbolic and logarithmic tangent integrals. For integers $m\geq n\geq1$ with $m+n$ even, the zeta-type formula \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n}(2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1} \] admits the beta-type analogue \[ \int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{\beta(2p)}{\pi^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m. \] The proof uses derivative polynomials, a beta kernel, and a coefficient-collapse argument. We then establish strict sign alternation for several coefficient families. In the boundary shifted-hyperbolic case the sign is controlled by generalized Bernoulli polynomials, while the arctanh and logarithmic tangent families are identified with continuous dual Hahn and continuous Hahn polynomials. Their zero distributions turn the observed alternation into structural statements valid for all admissible indices. Finally, with $\lambda(s)=(1-2^{-s})\zeta(s)$, we show that the parallel zeta--lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with $\varepsilon\in\{0,1\}$. The same convention $\varepsilon=0$ for the beta case and $\varepsilon=1$ for the zeta--lambda case is used throughout. Besides the shifted hyperbolic, odd-$\sinh$, logarithmic tangent, and $\tanh$ families, this framework yields a further unified pair of reciprocal-arctanh integral formulae on $(0,1)$.