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Congruences for \(\ell\)-regular partitions and bipartitions. (English) Zbl 1450.11108

The authors define the following \(q\)-series: \[ F(q)=\sum_{n=-\infty}^\infty(-1)^{\delta n}(an+b)q^{(cn^2+dn)/2}, \] where \(\delta, a, b, c, d\in\mathbb{Z}, c\neq0, c\equiv d\pmod{2}\), and \(a=2bc/d\neq0\) or \(a=0\). The function \(F(q)\) contains the Ramanujan general theta function \(f(a,b)=\sum\limits_{n=-\infty}^\infty a^{n(n-1)/2}b^{n(n+1)/2}\) \((|ab|<1)\) as a special case. Using standard \(q\)-series techniques, the authors derive an dissection identity for the coefficients of \(F(q)\) and use this identity to study congruence properties satisfied by the coefficients of \(F(q)\). As an immediate consequence, they obtain several infinite families of congruences for \(\ell\)-regular partition function and \(\ell\)-regular bipartition function. Moreover, based on the aforementioned dissection identity, the authors also give a new proof of the Ramanujan congruence for partition function modulo 5.

MSC:

11P83 Partitions; congruences and congruential restrictions
05A17 Combinatorial aspects of partitions of integers
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References:

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