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≥ n≥1 with m+n even, the zeta-type formula ∫0∞(tanhm+1x)/(xn+1) dx = (−1)(m−n)/2 ∑p=⌈ n/2⌉(m+n)/2 binom2pn(22p+1−1) (ζ(2p+1))/π2p [um+n−2p](ucot u)m+1 admits the beta-type analogue ∫0∞ (tanhm x)/(xncosh x) dx = (−1)(m−n)/2 ∑p=⌈ n/2⌉(m+n)/2 22pbinom2p−1n−1 β(2p)/(π2p−1) [um+n−2p] u/(sin u)(ucot 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 λ(s)=(1−2−s)ζ(s), we show that the parallel zeta--lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with ε∈{0,1}. The same convention ε=0 for the beta case and ε=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).
Related
- On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
- Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator
- Dirichlet Eigenvalues for the Logarithmic-Hardy operator
- Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space
- Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction
BibTeX
@misc{waffo2026dirichlet,
title = {{Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals}},
author = {Luc Ramsès Talla Waffo},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27717},
eprint = {2608.27717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27717},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals
AU - Luc Ramsès Talla Waffo
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27717
UR - https://arxiv.org/abs/2608.27717
AB - We develop a Dirichlet-beta counterpart and a structural extension of the coefficient-extraction framework for hyperbolic and logarithmic tangent integrals. For integers m≥ n≥1 with m+n even, the zeta-type formula ∫_0^∞(tanh^m+1x)/(x^n+1) dx = (−1)^(m−n)/2 ∑_p=⌈ n/2⌉^(m+n)/2 binom2pn(2^2p+1−1) (ζ(2p+1))/π^2p [u^m+n−2p](ucot u)^m+1 admits the beta-type analogue ∫_0^∞ (tanh^m x)/(x^ncosh x) dx = (−1)^(m−n)/2 ∑_p=⌈ n/2⌉^(m+n)/2 2^2pbinom2p−1n−1 β(2p)/(π^2p−1) [u^m+n−2p] u/(sin u)(ucot 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 λ(s)=(1−2^−s)ζ(s), we show that the parallel zeta--lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with ε∈{0,1}. The same convention ε=0 for the beta case and ε=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).
ER -