Vanishing Coefficients of q^{5n+r} and q^{7n+r} in Certain Infinite q-series Expansions

TL;DR

The paper investigates vanishing coefficients in arithmetic progressions of certain infinite $q$-series expansions, motivated by prior results on similar coefficient vanishing. It employs Ramanujan's theta function identities, the Jacobi triple product, and elementary $q$-series manipulations to rewrite products and reveal cancellations in structured sums, leading to explicit vanishing results. Key contributions include new vanishing identities modulo $5$ and $7$ for several infinite-product families (e.g., $a(5n+3)=0$, $b_2(5n+2)=0$, $e_2(7n+5)=0$, etc.), derived by decomposing expansions into components whose differences vanish. The work extends previous findings by Hirschhorn, Tang, and Ananya et al., providing a general framework for identifying vanishing coefficient patterns and suggesting avenues to explore periodicity in these $q$-series coefficients.

Abstract

Motivated by the recent work of several authors on vanishing coefficients of the arithmetic progression in certain $q$-series expansion, we study some variants of these $q$-series and prove some comparable results. For instance, if $\sum_{n=0}^{\infty}c_1(n)q^n=\left(\pm q^2,\pm q^3; q^5\right)_\infty^2 \left( q, q^{14}; q^{15}\right)_\infty$, then $c_1(5n+3)=0$.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• proof : Proof
• Remark 3.2
• Theorem 3.3
• proof : Proof
• Theorem 3.4