## 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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