THE RAMANUJAN JOURNAL, Vol. 6, No. 1 (2002), 7-149.

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

STEPHEN C. MILNE
Department of Mathematics, The Ohio State University, Columbus, Ohio, 43210
milne@math.ohio-state.edu

Received August 5, 2000; Accepted March 1, 2001

Abstract. 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. Our $24$ squares identity leads to a different formula for Ramanujan's tau function $\tau (n)$, when $n$ is odd. 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 Turánian 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. Our $16$ and $24$ squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's $C_{\ell}$ nonterminating ${}_6\phi_5$ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's $2$, $4$, $6$, 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 $n^2$ and $n(n+1)$ squares. Our sums of more than $8$ squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared.

Keywords: Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, Eisenstein series, inclusion/exclusion, Laplace expansion formula for determinants, Schur functions, multiple basic hypergeometric series, $C_{\ell}$ nonterminating $ _6\phi_5$ summation theorem, lattice sums

2000 Mathematics Classification: Primary 11E25, 33E05; Secondary 05A15, 33D70