Infinite product representations of some q-series

Florian Münkel, Lerna Pehlivan, Kenneth S. Williams*

*Corresponding author for this work

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For integers $a$ and $b$ (not both $0$) we define the integers $c(a,b;n)\ \ (n=0,1,2,\ldots)$ 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 $t_k(n) = c(-k,2k;n)$, which count the number of representations of $n$ as a sum of $k$ triangular numbers, and the numbers $(-1)^n r_k(n) = c(2k,-k;n)$, where $r_k(n)$ counts the number of representations of $n$ as a sum of $k$ squares. A computer search was carried out for integers $a$ and $b$, satisfying $-24\leq a,b\leq 24$, 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 $q$ and a single infinite product of factors involving powers of $1-q^{rn}$ with $r\in\{1,2,3,4,6,8,12,24\}$ for all powers of $q$ up to $q^{1000}$. A total of 84 such candidate identities involving 56 pairs of integers $(a,b)$ 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 \[\sum_{n=0}^{infty} t_k(3n+j)q^n, \quad \sum_{n=0}^{infty} r_k(3n+j)q^n, \quad j=0,1,2. \]
Original languageEnglish
Pages (from-to)839-872
Number of pages34
JournalRamanujan Journal
Early online date16 Oct 2023
Publication statusPublished - Mar 2024


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