A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup Γ 0 (2) of the modular group SL 2 (Z) is constructed. These nonlinear equations are analogues of the well known Ramanujan equations, as well as the Chazy and Darboux-Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz triangle function S(0, 0, 1/2; z).

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums σ s (n) = d|n (−1) d−1 d s and σ s (n) = d|n (−1) n d −1 d s , for certain s, which were first defined by Glaisher. We first introduce three functions P(q), E(q), and Q(q) related to σ(n), σ(n), and σ 3 (n), respectively, and then we evaluate them in terms of two parameters x and z in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining r s (n) and δ s (n), s = 4, 8, in terms of σ(n), σ(n), and σ 3 (n), where r s (n) denotes the number of representations of n as a sum of s squares and δ s (n) denotes the number of representations of n as a sum of s triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

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In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands' beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of Sym j pSym k V q for GL 2 in the case k " 3. We compute the plethysm explicitly for the cases k " 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of GL 2 for these two cases.

In this paper, we define the normalized Eisenstein series P, e, and Q associated with 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 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.