Steve Milne

Affiliation: Ohio State University

Email: milne@math.ohio-state.edu

Title Of Talk: Sums of squares, Schur functions, and multiple basic hypergeometric series

Abstract: Sums of squares, Schur functions, and multiple basic hypergeometric series We start with a brief review of the sums of squares problem from Jacobi, Glaisher, and Ramanujan, on to the present, followed by a quick summary of one-variable basic hypergeometric series. We then give some necessary background on multiple basic hypergeometric series and their connection to the sums of squares problem. Motivated by Andrews' (one-variable) basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities, we discuss how we used multiple basic hypergeometric series, Gustafson's C_l nonterminating ₆φ₅ summation theorem, and symmetry and Schur function techniques to prove the existence of three basic types of infinite families of explicit exact non-trivial closed formulas for the number of ways of writing a positive integer N as a sum of a given number of squares of integers, without using coefficients of cusp forms. These three types of infinite families involve either 4n² or 4n(n+1) squares, 2n(2n-1) or 2n(2n+1) squares, or n² or n(n+1) squares, respectively. The n=1 case is classical. We first computed the explicit n=2, and/or n=3 cases by the aid of Mathematica. All of these identities, even for n=2 or n=3, for example 9, 12, and 20 squares, are of significant interest. We briefly note how we subsequently used combinatorial/elliptic function methods to actually derive these explicit exact non-trivial closed formulas, from our first type of infinite families, for 4n² or 4n(n+1) squares of integers, respectively. Similar combinatorial/elliptic function methods allow us to recover the n=2 case of our 2n(2n-1) or 2n(2n+1) squares identities. These 12 and 20 squares formulas, which extend Jacobi's 2 and 6 squares identities, respectively, are in the form of 2 by 2 determinants of Lambert series that expand θ₃(0,-q) to the 12-th and 20-th power, respectively. These are not Hankel determinants as in the 4n² or 4n(n+1) squares identities, but a subtle variation. Our methods should also work for the general 2n(2n-1) or 2n(2n+1) squares case, yielding deep extensions of Hankel determinants. In addition to the three types of infinite families of sums of squares identities discussed above, we are also able to obtain the product sides of many infinite families of Macdonald's η-function identities.

Last update made Fri Mar 5 18:23:00 EST 2010.
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