Vanishing coefficient results in four families of infinite q-products
S. Ananya, Channabasavayya, D. Ranganatha, R. G. Veeresha
TL;DR
The paper investigates vanishing coefficients in four families of infinite $q$-products defined by $X_{a,b,s\ell,k\ell,u,v}(n)$, $Y_{a,b,s\ell,k\ell,u,v}(n)$, $Z_{a,b,s\ell,k\ell,u,v}(n)$, and $W_{a,b,s\ell,k\ell,u,v}(n)$, and proves that coefficients vanish in arithmetic progressions modulo $5$, $7$, $11$, $13$, $19$, $21$, $23$, and $29$. It advances the Ramanujan–Richmond–Szekeres–Andrews lineage by applying Jacobi triple product identities and Ramanujan's general theta function to derive numerous residue-class vanishings, facilitated by the extraction operator $E_{k,l}$ and index transformations. The central technique involves expressing the four products via $f(a,b)$-type theta functions and systematically extracting $(kn+l)$-components to show zeros in specified congruence classes, with several explicit identities such as $X_{t,2t,5\ell,15\ell,2,1}(5n+2t)=0$ and related cases. These results expand the catalog of vanishing coefficient phenomena in infinite $q$-products and point to broader generalizations for other parameter choices.
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 $\{X_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$, $\{Y_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$, $\{Z_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$ and $\{W_{a,b,sm,km,u,v}(n) \}_{n\geq n_0}$ 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.