Dirichlet Beta 类似物、符号交替及双曲与对数正切积分的 Lerch 统一
Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals
AI总结：
该研究开发了双曲与对数正切积分的 Dirichlet-beta 对应物及系数提取框架，建立了相关系数族的符号交替性，通过 Lerch 超越函数统一了 zeta 与 beta 积分恒等式。
AI中文摘要：
我们开发了 Dirichlet-beta 对应物以及双曲和对数正切积分的系数提取框架的结构扩展。对于满足 $m\geq n\geq1$ 且 $m+n$ 为偶数的整数，zeta 型公式 $\int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n}(2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1}$ 存在 beta 型类似物 $\int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{\beta(2p)}{\pi^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m$。证明使用了导数多项式、beta 核和系数坍缩论证。随后，我们为多个系数族建立了严格的符号交替性：在边界移位双曲情形中，符号由广义伯努利多项式控制；而反正切和对数正切族分别与连续双 Hahn 多项式和连续 Hahn 多项式等同，它们的零点分布将观测到的交替转化为对所有容许指标成立的结构性陈述。最后，令 $\lambda(s)=(1-2^{-s})\zeta(s)$，我们证明平行的 zeta–lambda 和 beta 恒等式是单个 Lerch 超越函数方案的两个二元特化，其中 $\varepsilon\in\{0,1\}$，全程采用 beta 情形对应 $\varepsilon=0$、zeta–lambda 情形对应 $\varepsilon=1$ 的约定。除移位双曲、奇数 sinh、对数正切和 tanh 族外，该框架还在 $(0,1)$ 上产生了另一对互反反正切积分公式。
英文摘要：
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 \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n}(2^{2p+1}-1) \frac{ζ(2p+1)}{π^{2p}} [u^{m+n-2p}](u\cot u)^{m+1} \] admits the beta-type analogue \[ \int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{β(2p)}{π^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m. \] 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 $λ(s)=(1-2^{-s})ζ(s)$, we show that the parallel zeta--lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with $\varepsilon\in\{0,1\}$. 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 $(0,1)$.
