A polynomial method for proving congruences of pω(n,m) and pν(n,m)

Abstract

Recently, two partition functions [Formula: see text] and [Formula: see text] were introduced by Andrews et al. The partition function [Formula: see text] denotes the number of partitions of [Formula: see text] in which each odd part is less than twice the smallest part, and [Formula: see text] counts the number of partitions of [Formula: see text] into distinct non-negative parts such that all odd parts are less than twice the smallest part. Very recently, Silva et al. studied congruence properties of the restricted partition functions [Formula: see text] and [Formula: see text] which denote the number of partitions enumerated by [Formula: see text] and [Formula: see text], respectively, into exactly [Formula: see text] parts. In this paper, we give a polynomial method for discovering congruences for [Formula: see text] and [Formula: see text] by checking a finite number of initial values. Employing the polynomial method, we proved new and existing congruences for [Formula: see text] and [Formula: see text] based on a bounded number of calculations. For example, in order to prove that [Formula: see text] holds for all [Formula: see text], it suffices to show that the congruence holds when [Formula: see text].