交错赫维茨zeta函数的拉马努金型恒等式
Ramanujan-type identities for alternating Hurwitz zeta functions
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中文总结 AI 辅助
将拉马努金关于\(\zeta(2n + 1)\)的恒等式扩展到交错赫维茨zeta函数，研究其在不同模对称条件下性质及对应恒等式，还建立相关函数无穷级数表达式与变换公式。
中文摘要 AI 辅助
约1910年，拉马努金在手稿中为\(\zeta(2n + 1)\)提出恒等式。本文将其扩展到交错赫维茨zeta函数，系统研究其性质及恒等式，建立正切和双曲正切函数乘积的无穷级数表达式，定义奇偶阶交错赫维茨核并获相关恒等式及变换公式。
英文摘要
Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $ζ(2n+1)$: \[ \begin{aligned} α^{-n}\,&\left\{\dfrac{1}{2}\,ζ(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2αm} - 1}\right\} \\ &\quad\quad\quad\quad\quad\quad\quad-(-β)^{-n}\,\left\{\dfrac{1}{2}\,ζ(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2βm} - 1}\right\}\\ &=2^{2n}\sum_{k = 0}^{n + 1}\dfrac{(-1)^{k-1}B_{2k}\,B_{2n - 2k + 2}}{(2k)!(2n - 2k + 2)!}\,α^{n - k + 1}β^k, \end{aligned} \] where $α$, $β$ are positive numbers satisfying $αβ=π^2,n\in\mathbb{N},$ $B_n$ denotes the $n$-th Bernoulli number and $ζ(z)$ is the Riemann zeta function. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler's famous formula for even zeta values. In this paper, we extend Ramanujan's identity to the alternating Hurwitz zeta function. Then we systematically investigate the properties of the alternating Hurwitz zeta function $ζ_E(z,x)$, as well as the corresponding Ramanujan-type identities, under different modular symmetry conditions. We also establish infinite series expressions for products of the tangent and hyperbolic tangent functions, and express the Dirichlet lambda function $λ(z)$ together with linear combinations of infinite series as convolution sums of special sequences. Furthermore, we define alternating Hurwitz kernels of even and odd orders, and obtain Ramanujan-type identities involving the alternating digamma function $\widetildeψ(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels.
