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Abstract: In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order. [Copyright &y& Elsevier]

Abstract: It is conjectured by Erdős, Graham and Spencer that if with , then this sum can be decomposed into n parts so that all partial sums are ⩽1. This is not true for as shown by , , . In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for . In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for . Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if and no partial sum (certainly not a single term) is the inverse of an positive integer. [Copyright &y& Elsevier]

Abstract: The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition. [Copyright &y& Elsevier]