Congruences for Fourier coefficients of eta‐quotients modulo powers of 5, 7, 11, 13, and 17

In this paper, we investigate the Fourier coefficients of eta-quotients forms q λ 0 + l k 1 24 η ( τ ) − , $$ {q}^{\frac{\lambda_0+{l}^k{\lambda}_1}{24}}\eta {\left(\tau \right)}^{-{\lambda}_0}\eta {\left({l}^k\tau \right)}^{-{\lambda}_1}, where \eta \left(\tau \right) is Dedekind eta function, = 5,7 11 13 l=5,7,11,13 and 17; a positive integer, {\lambda}_0,{\lambda}_1 are arbitrary integers. We prove Ramanujan's type congruences for \right)}^{-{\lambda}_1} modulo powers prime . recover several results due to Atkin, Garvan, Gordon, Wang, Mestrige, others. give few examples, establish an improvement Wang's related 11-regular partition function.