Ramanujan-type identities for alternating Hurwitz zeta functions

Abstract

Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $ζ(2n+1)$: \begin{align*} α^{-n}\left\{\frac{1}{2}ζ\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2αm}-1}\right\} &-\left(-β\right)^{-n}\left\{\frac{1}{2}ζ\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2βm}-1}\right\} \\&=2^{2n}\sum_{k=0}^{n+1}{\frac{\left(-1\right)^{k-1}B_{2k}B_{2n-2k+2}} {\left(2k\right)!\left(2n-2k+2\right)!}α^{n-k+1}β^k}, \end{align*} where $α$, $β$ are positive numbers satisfying $αβ=π^2,n\in\mathbb Z\setminus\{0\},$ $B_n$ denotes the $n$-th Bernoulli number, and $ζ(z)$ is the Riemann zeta function. In this paper, we extend Ramanujan's identity to the alternating Hurwitz zeta function and systematically investigate the properties of the alternating Hurwitz zeta function $ζ_E(z,x)$ under different modular symmetry conditions, as well as the corresponding Ramanujan-type identities. 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.