NEW CONGRUENCES AND DENSITY RESULTS FOR t-REGULAR PARTITIONS WITH DISTINCT EVEN PARTS

Abstract

Let $t\ge 2$ be a fixed positive integer. Let ${ped}_t(n)$ denote the number of $t$-regular partitions of $n$ wherein the even parts are distinct and the odd parts are unrestricted. We establish infinite families of congruences for ${ped}_t(n)$ modulo certain positive integers $M$, for specific values of $t$. We next study the distribution of ${ped}_t(n)$ for $t=3,5,7,9$. We prove that the series ${\sum }_{n=0}^{\infty }{ped}_t(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$ for $t=3,5,9$. We also prove that the series ${\sum }_{n=0}^{\infty }{ped}_7(2n+1)q^n$ is lacunary modulo $2$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.