Eisenstein series associated with $\Gamma_0(2)$

TL;DR

This paper studies Eisenstein series related to a0a0a0a0a0a0a0a0a0a0a0(2) and explores their differential equations, modular solutions, and connections to combinatorics and special series.

Contribution

It introduces new Eisenstein series for a0a0a0a0a0a0a0a0a0a0a0(2), derives associated differential equations, and constructs solutions using orthogonal polynomials and hypergeometric series.

Findings

Derived three differential equations for Eisenstein series.

Constructed modular solutions using orthogonal polynomials and hypergeometric series.

Established a combinatorial identity involving triangular numbers.

Abstract

In this paper, we define the normalized Eisenstein series $\mathcal{P}$, $e$, and $\mathcal{Q}$ associated with $\Gamma_0(2),$ 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 $\Gamma_0(2)$ 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.