Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions

Details Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions

2000-08-09

arxiv.org/abs/math/0008068v2

The Ramanujan Journal 6 (2002), 7--149. (Preface by George Andrews on pages 5--6.) Also in book as vol. 5 of ``Developments in

Mathematics

Number Theory

AmSTeX file of final 12-31-00 preprint version. Prints 125 pages. To appear as a special issue of The Ramanujan Journal

Scientific paper

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n^2 or 4n(n+1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turanian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the eta-function identities of Macdonald. Moreover, the powers 4n(n+1), 2n^2+n, 2n^2-n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac--Wakimoto conjectured identities involving representing a positive integer by sums of 4n^2 or 4n(n+1) triangular numbers, respectively.

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