Summary: In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to or squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function , when 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 -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 or triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s nonterminating 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 or 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.