Arithmetic properties of $k$-tuple $\ell$-regular partitions

Abstract

In this paper, we study arithmetic properties satisfied by the $k$-tuple $\ell$-regular partitions. A $k$-tuple of partitions $(\hat{I}{¾}_1,\hat{I}{¾}_2,\dots ,\hat{I}{¾}_k)$ is said to be $\ell$-regular if all the $\hat{I}{¾}_i$'s are $\ell$-regular. We study the cases $(\ell ,k)=(2,3),(4,3),(\ell ,p)$, where $p$ is a prime, and even the general case when both $\ell$ and $k$ are unrestricted. Using elementary means as well as the theory of modular forms we prove several infinite family of congruences and density results for these family of partitions.

22 pages, minor changes as per referee suggestions, to appear in the Journal of Mathematical Analysis and Applications