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

TL;DR

This paper establishes new internal congruences modulo powers of 3 for functions derived from Ramanujan's theta functions, extending previous results and providing deeper insights into their modular properties.

Contribution

It introduces generalized internal congruences for functions from Ramanujan's theta functions modulo arbitrary powers of 3, expanding upon prior specific cases.

Findings

Proves congruences for ph_3(n) involving powers of 3.

Establishes congruences for ps_3(n) with shifted arguments.

Generalizes earlier results by Bharadwaj et al. and Gireesh et al.

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.