Eisenstein series associated with Γ0(2)

Abstract

In this paper, we define the normalized Eisenstein series ℘, e, and $\mathcal{Q}$ associated with Γ0(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ0(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.