Ramanujan's theta functions and internal congruences modulo arbitrary powers of 3

Abstract

In this work, we investigate internal congruences modulo arbitrary powers of 3 for two functions arising from Ramanujan's classical theta functions φ(q) and ψ(q). By letting∑n≥0ph3(n)qn:=φ(−q3)φ(−q)and∑n≥0ps3(n)qn:=ψ(q3)ψ(q), we prove that for any m≥1 and n≥0,ph3(32m−1n)≡ph3(32m+1n)(mod3m+2), andps3(32m−1n+32m−14)≡ps3(32m+1n+32m+2−14)(mod3m+2), thereby substantially generalizing the previous results of Bharadwaj et al. and Gireesh et al., respectively.