Vanishing coefficient results in four families of infinite q-products

Abstract

In the recent past, the work in the area of vanishing coefficients of infinite $q$-products has been taken to the forefront. Weaving the same thread as Ramanujan, Richmond, Szekeres, Andrews, Alladi, Gordon, Mc Laughlin, Baruah, Kaur, Tang, we further prove vanishing coefficients in arithmetic progressions moduli 5, 7, 11, 13, 19, 21, 23 and 29 of the following four families of infinite products, where $\{X_{a,b,sm,km,u,v}(n){\}}_{n\ge n_0}$, $\{Y_{a,b,sm,km,u,v}(n){\}}_{n\ge n_0}$, $\{Z_{a,b,sm,km,u,v}(n){\}}_{n\ge n_0}$ and $\{W_{a,b,sm,km,u,v}(n){\}}_{n\ge n_0}$ are defined by \begin{align*} \sum_{n\geq n_0}^{\infty}X_{a,b,sm,km,u,v}(n)q^n:=&(q^{a},q^{sm-a};q^{sm})_{infty}^u(q^{b},q^{km-b};q^{km})_{infty}^v, \\ \sum_{n\geq n_0}^{\infty}Y_{a,b,sm,km,u,v}(n)q^n:=&(q^{a},q^{sm-a};q^{sm})_{infty}^u(-q^{b},-q^{km-b};q^{km})_{infty}^v, \\ \sum_{n\geq n_0}^{\infty}Z_{a,b,sm,km,u,v}(n)q^n:=&(-q^{a},-q^{sm-a};q^{sm})_{infty}^u(q^{b},q^{km-b};q^{km})_{infty}^v,\\ \sum_{n\geq n_0}^{\infty}W_{a,b,sm,km,u,v}(n)q^n:=&(-q^{a},-q^{sm-a};q^{sm})_{infty}^u(-q^{b},-q^{km-b};q^{km})_{infty}^v, \end{align*} here $a,b,s,k,u$ and $v$ are chosen in such a way that the infinite products in the right-hand side of the above are convergent and $n_0$ is an integer (possibly negative or zero) depending on $a,b,s,k,u$ and $v$. The proof uses the Jacobi triple product identity and the properties of Ramanujan general theta function.