2 research outputs found
Infinite product representations of some q-series
For integers and (not both ) we define the integers by \sum_{n=0}^{infty} c(a,b,;n)q^n = \prod_{n=1}^\infty \left(1-q^n\right)^a (1-q^{2n})^b \quad (|q|<1). These integers include the numbers , which count the number of representations of as a sum of triangular numbers, and the numbers , where counts the number of representations of as a sum of squares. A computer search was carried out for integers and , satisfying , such that at least one of the sums \begin{align} \sum_{n=0}^{infty} c(a,b;3n+j)q^n, \quad j=0,1,2, \end{align} (0.1) is either zero or can be expressed as a nonzero constant multiple of the product of a power of and a single infinite product of factors involving powers of with for all powers of up to . A total of 84 such candidate identities involving 56 pairs of integers all satisfying a\equiv b\pmd3 were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums \[\sum_{n=0}^{infty} t_k(3n+j)q^n, \quad \sum_{n=0}^{infty} r_k(3n+j)q^n, \quad j=0,1,2. \