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Abstract
We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.
Highlights
Introduction and statement of resultsAn overpartition of n is a partition of n in which the final occurrence of a number may be overlined
We use q-difference equations to compute a two-variable q-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form
Let s(n) denote the number of overpartitions counted by t(n) but with odd smallest part
Summary
An overpartition of n is a partition of n in which the final occurrence of a number may be overlined. Equation (6) combined with work of the first and the third author on overpartitions and class numbers [6, 7] implies that the generating function for t+(n) − t−(n) is an eigenform modulo 3 for the weight 3/2 Hecke operators This is recorded below along with a congruence for t(n) modulo 3. Since (5) and (7) are (up to rational q-powers) weakly holomorphic modular forms of non-positive weight, Rademacher and Zuckerman’s famous refinement of the Hardy-Ramanujan Circle Method applies [9, 12, 13] These results allow one to use the cuspidal principal parts in order to calculate exact formulas for the coefficients.
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