New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function

摘要

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 Turanian determinants, Fourier series, Lambert series, inclusion/ exclusion, Laplace expansion formula for determinants, and Schur functions.