Arithmetic properties derived from coefficients of certain eta quotients

Abstract

Abstract For a positive integer k, let F ( q ) k : = ∏ n ≥ 1 ( 1 − q n ) 4 k ( 1 + q 2 n ) 2 k = ∑ n ≥ 0 a k ( n ) q n $$ F (q)^{k}:= \prod_{n \geq 1} \frac{(1-q^{n})^{4k}}{(1+q^{2n})^{2k}} = \sum_{n\geq 0} \frak{a}_{k} (n)q^{n} $$ be the eta quotients. The coefficients a 1 ( n ) $\frak{a}_{1} (n)$ can be interpreted as a certain kind of restricted divisor sums. In this paper, we give the signs and modulo values for a 1 ( n ) $\frak{a}_{1} (n)$ and a 2 ( m ) $\frak{a}_{2} (m)$ and calculate several convolution sums involving a k ( n ) $\frak{a}_{k} (n)$ .

Keywords

• Restricted divisor functions
• Eta quotient
• Convolution sums
• q-series