Ramanujan's theta functions and internal congruences modulo arbitrary powers of 3 (Q6635207)

Let \(\varphi(q)\) denote the classical theta function, \N\[\N\varphi(q) = \sum_{n \in \mathbb{Z}} q^{n^2}, \N\]\Nand define \(ph_3(n)\) via the \(q\)-expansion \N\[\N\sum_{n \geq 0} ph_3(n)q^n = \frac{\varphi(-q^3)}{\varphi(-q)}.\N\]\NThe authors prove the family of congruences \N\[\Nph_3(3^{2m+1}n) \equiv ph_3(3^{2m-1}n) \pmod{3^{m+2}}, \N\]\Nvalid for all \(m \geq 1\) and \(n \geq 0\). To prove this result they express the series \N\[\N\frac{\varphi(-q^3)}{\varphi(-q^9)} \sum_{n \geq 0} (ph_3(3^{2m+1}n) - ph_3(3^{2m-1}n))q^n\N\]\Nas a polynomial in \N\[\N\xi = \frac{\varphi(-q^9)}{\varphi(-q)}\N\]\Nwith integer coefficients, and they show that these coefficients are all divisible by the necessary powers of \(3\). A key role is played by the \(U_3\)-operator acting on modular functions of level \(18\). The authors also establish a similar result for the coefficients of \N\[\N\frac{\psi(q^3)}{\psi(q)},\N\]\Nwhere \(\psi(q)\) is the theta function \N\[\N\psi(q) = \sum_{n \geq 0}q^{\binom{n+1}{2}}.\N\]