Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities

Abstract

Recently, Lin and Wang introduced two special partition functions $$RG_1(n)$$ and $$RG_2(n)$$ , the generating functions of which are the reciprocals of two identities due to Ramanujan and Gordon. They established several congruences modulo 5 and 7 for $$RG_1(n)$$ and $$RG_2(n)$$ and posed four conjectures on congruences modulo 25 for $$RG_1(n)$$ and $$RG_2(n)$$ at the end of their paper. In this paper, we confirm the four conjectures given by Lin and Wang by using Ramanujan’s modular equation of fifth degree. Moreover, we also obtain new congruences modulo 25 for $$RG_1(n)$$ and $$RG_2(n)$$ based on Newman’s identities. For example, we deduce that for any integer $$n\ge 0$$ , $$\begin{aligned} RG_1\left( \frac{23375n(3n+1)}{2}+974\right)&\equiv RG_1\left( \frac{23375n(3n+5)}{2}+24349\right) \\&\equiv 0\pmod {25}. \end{aligned}$$