Congruences modulo prime for fractional colour partition function

Abstract

PurposeLet p[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.Design/methodology/approachUsing q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.FindingsIn the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.Originality/valueThe partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.