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 $\varphi(q)$ and $\psi(q)$. By letting \begin{align*} \sum_{n\ge 0} ph_3(n) q^n:=\dfrac{\varphi(-q^3)}{\varphi(-q)}\qquad\text{and}\qquad \sum_{n\ge 0} ps_3(n) q^n:=\dfrac{\psi(q^3)}{\psi(q)}, \end{align*} we prove that for any $m\ge 1$ and $n\ge 0$, \begin{align*} ph_3\big(3^{2m-1}n\big)\equiv ph_3\big(3^{2m+1}n\big)\pmod{3^{m+2}}, \end{align*} and \begin{align*} ps_3{\left(3^{2m-1}n+\frac{3^{2m}-1}{4}\right)}\equiv ps_3{\left(3^{2m+1}n+\frac{3^{2m+2}-1}{4}\right)}\pmod{3^{m+2}}, \end{align*} thereby substantially generalizing the previous results of Bharadwaj et al.~and Gireesh et al., respectively.