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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 $4{n}^{2}$ or $4n\left(n+1\right)$ squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function $\tau \left(n\right)$, 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 $\eta$-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 $4{n}^{2}$ or $4n\left(n+1\right)$ triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s ${C}_{\ell }$ nonterminating ${}_{6}{\varphi }_{5}$ 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 $n\left(n+1\right)$ 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.

##### MSC:
 11P82 Analytic theory of partitions 11F11 Holomorphic modular forms of integral weight 33E05 Elliptic functions and integrals