Arithmetic of General Partition Functions $p_r(n)$ Modulo Primes

Abstract

In the present investigation, we establish several new infinite families of congruences for the generalized partition function $p_r(n)$. Our emphasis throughout this paper is on demonstrating how classical and modern $q$-identities can be effectively employed to derive these congruences. By systematically applying these identities, we uncover congruences modulo primes such as 19, 23 and 29, valid for all positive integers $\lambda$. This approach not only yields elegant arithmetic results but also highlights the deep interplay between partition theory and the analytic properties of $q$-series.