Ramanujan-type identities for alternating Hurwitz zeta functions

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The paper extends a Ramanujan identity for the Riemann zeta function to the alternating Hurwitz zeta function, exploring its properties under modular symmetry conditions and establishing new Ramanujan-type identities. This extension is significant as it broadens the applicability of Ramanujan's work to more complex zeta functions and related special functions. A concrete detail includes the derivation of infinite series expressions for products involving tangent and hyperbolic tangent functions, connected to convolution sums of specific sequences.

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arXiv:2607.03490v1 Announce Type: new Abstract: Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $\zeta(2n+1)$: \begin{align*} \alpha^{-n}\left\{\frac{1}{2}\zeta\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2\alpha m}-1}\right\} &-\left(-\beta\right)^{-n}\left\{\frac{1}{2}\zeta\left(2n+1\right) +\sum_{m=1}^{\infty}\frac{m^{-2n-1}}{e^{2\beta m}-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)!}\alpha^{n-k+1}\beta^k}, \end{align*} where $\alpha$, $\beta$ are positive numbers satisfying $\alpha\beta=\pi^2,n\in\mathbb Z\setminus\{0\},$ $B_n$ denotes the $n$-th Bernoulli number, and $\zeta(z)$ 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 $\zeta_E(z,x)$ 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 $\lambda(z)$ 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 $\widetilde{\psi}(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels.