On generalized Eisenstein series and Ramanujan's formula for periodic zeta-functions

Abstract: In this paper, transformation formulas for a large class of Eisenstein series defined by [ G(z,s;A_{\alpha},B_{\beta};r_{1},r_{2})=\sum\limits_{m,n=-\infty}^{\infty }\ \hspace{-0.19in}^{^{\prime}}\frac{f(\alpha m)f^{\ast}(\beta n)} {((m+r_{1})z+n+r_{2})^{s}},\text{ }\operatorname{Re}(s)>2,\text{ }\operatorname{Im}(z)>0 ] are investigated for $s=1-r$, $r\in\mathbb{N}$. Here $\left{ f(n)\right}$ and $\left{ f^{\ast}(n)\right}$, $-\infty<n<\infty$ are sequences of complex numbers with period $k\>0$, and $A_{\alpha}=\left{ f(\alpha n)\right} $ and $B_{\beta}=\left{ f^{\ast}(\beta n)\right}$, $\alpha,\beta\in\mathbb{Z}$. Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol-Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $G(z,s;A_{\alpha},I;r_{1},r_{2})$ and $G(z,s;I,A_{\alpha };r_{1},r_{2})$, where $I=\left{ 1\right}$. As an application of these formulas, analogues of Ramanujan's formula for periodic zeta-functions are derived.