Congruences for modular forms and applications to crank functions

TL;DR

This paper establishes new congruences for a broad class of modular forms, generalizes crank generating functions, and reveals connections between various partition statistics and the partition function.

Contribution

It introduces new congruences for modular forms, generalizes crank generating functions, and links different partition statistics to the partition function.

Findings

Derived congruences for modular forms.

Generalized the crank generating function.

Connected birank and k-crank to the partition function.

Abstract

In this paper, motivated by the work of Mahlburg, we find congruences for a large class of modular forms. Moreover, we generalize the generating function of the Andrews-Garvan-Dyson crank on partition and establish several new infinite families of congruences. In this framework, we showed that both the birank of an ordered pair of partitions introduced by Hammond and Lewis, and $k$-crank of $k$-colored partition introduced by Fu and Tang process the same as the partition function and crank.