Ramanujan-type identities for alternating Hurwitz zeta functions

Abstract

Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $\zeta (2n+1)$: \[ \begin{aligned} α^{-n}\,&\left\{\dfrac{1}{2}\,ζ(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2αm} - 1}\right\} \\ &\quad\quad\quad\quad\quad\quad\quad-(-β)^{-n}\,\left\{\dfrac{1}{2}\,ζ(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2βm} - 1}\right\}\\ &=2^{2n}\sum_{k = 0}^{n + 1}\dfrac{(-1)^{k-1}B_{2k}\,B_{2n - 2k + 2}}{(2k)!(2n - 2k + 2)!}\,α^{n - k + 1}β^k, \end{aligned} \] where $\alpha$, $\beta$ are positive numbers satisfying $\alpha \beta ={\pi }^2,n\in \mathbb{N},$ $B_n$ denotes the $n$-th Bernoulli number and $\zeta (z)$ 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 ${\zeta }_E(z,x)$, 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 $\lambda (z)$ 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 $\tilde{\psi }(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels.

48 pages