Analogues of Ramanujan’s partition identities

Abstract

Ramanujan discovered that

$$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$

where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by

$$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$

They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$.