Affiliation: Ohio State University
Email: milne@math.ohio-state.edu
Title Of Talk: New Lambert series formulas for 12 and 20 squares, and multiple basic hypergeometric series
Abstract: We first discuss how basic hypergeometric series in one and several variables lead to formulas for sums of squares, including our recent work on infinite families of such formulas. After a review of our formulas for $16$ and $24$ squares, we present our new expansion of $\vartheta_3(0,-q)^{12}$ and $\vartheta_3(0,-q)^{20}$ as $2$ by $2$ determinants of double power series, where $\vartheta_3(0,q)$ is the classical theta function with $j$-th term $q^{j^2}$. We then express the double power series involved as linear combinations of classical Lambert series. The resulting Lambert series expansions here, as well as our earlier analogous $2$ by $2$ determinant expansions of $\vartheta_3(0,-q)^{16}$ and $\vartheta_3(0,-q)^{24}$, directly extend (and contain) Jacobi's classical formulas for $2$, $4$, $6$, and $8$ squares to $12$, $16$, $20$, and $24$ squares, respectively.
Last update made Thu Nov 11 14:46:57 EST 2004.
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