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2010•Proceedings of the American Mathematical SocietyOpen access

Congruences via modular forms

Robert Osburn, Brundaban Sahu

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Abstract

We prove two congruences for the coefficients of power series expansions in t t of modular forms where t t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

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We prove two congruences for the coefficients of power series expansions in t t of modular forms where t t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

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Available abstract

We prove two congruences for the coefficients of power series expansions in t t of modular forms where t t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

Key concepts: Congruence relation, Modular form, Modular design, Mathematics, Power series, Formal power series, Generating function, Series (stratigraphy)

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