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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. (English) Zbl 1125.11346

Summary: In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n 2 or 4n(n+1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turán determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of I. G. Macdonald ’s work [Invent. Math. 15, 91–143 (1972; Zbl 0244.17005)]. A special case of our methods yields a proof of the two conjectured [V. G. Kac and M. Wakimoto, Prog. Math. 123, 415–456 (1994; Zbl 0854.17028)] identities involving representing a positive integer by sums of 4n 2 or 4n(n+1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s C nonterminating 6 ϕ 5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities.

We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n 2 or n(n+1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

11P82Analytic theory of partitions
11F11Holomorphic modular forms of integral weight
33E05Elliptic functions and integrals