Vanishing coefficients in arithmetic progressions of some infinite products with modulo 13, 17, 19 and 29

Abstract

Quite recently, Tang extensively studied the vanishing coefficients in arithmetic progressions of the following three types of infinite q-products:

$$\begin{aligned} \sum _{n=n_0}^{\infty } \epsilon _{j, k, r, s, u}(n)q^n&=\left( q^j,q^{r-j};q^r\right) _{\infty }^s\left( q^k,q^{2r-k};q^{2r}\right) _{\infty }^u, \\ \sum _{n=n_0}^{\infty } \gamma _{j, k, r, s, u}(n)q^n&=\left( -q^j,-q^{r-j};q^r\right) _{\infty }^s\left( q^k,q^{2r-k};q^{2r}\right) _{\infty }^u, \\ \sum _{n=n_0}^{\infty } \delta _{j, k, r, s, u}(n)q^n&=\left( q^j,q^{r-j};q^r\right) _{\infty }^s\left( -q^k,-q^{2r-k};q^{2r}\right) _{\infty }^u, \end{aligned}$$

where $j, k, s, u\in {\mathbb {Z}}^{+}$, $r\ge 2$, and $n_0$ is an integer (possibly negative or zero) depending on j, k, r, s, u. In this paper, we further explore the vanishing coefficients for arithmetic progressions modulo 13, 17, 19 and 29 in the above three infinite products. For example, we prove that

$$\begin{aligned} \epsilon _{2t,t,13\alpha ,3,2}(13n+4t)=\epsilon _{3t,8t,17\alpha ,4,1} (17n+10t)=\epsilon _{2t,5t,29\alpha ,1,2}(29n+6t)=0, \end{aligned}$$

where $\alpha \ge 1$, $\gcd (r,t)=1$ with $r \in \{13,17,19,29\}$.