On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type

In this paper, we consider certain analogues of Eisenstein seriesinvolving hyperbolic sine and cosine functions. Using a resultof Hurwitz, we prove some evaluation formulas for these seriesat positive integers. We can regard these formulas as analoguesof the famous formulas for certain series given by Cauchy, Ramanujan,Berndt, and so on, as well as those for the Eisenstein seriesgiven by Hurwitz.