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Zbl 1125.11315
Milne, Stephen C.
Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. (English)
[J] Ramanujan J. 6, No. 1, 7-149 (2002). ISSN 1382-4090; ISSN 1572-9303/e

Summary: 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. 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 Turán 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. The article has also been published in monograph form [Developments in Mathematics 5. Boston MA: Kluwer Academic Publishers (2002; Zbl 1125.11316)]. An announcement appeared in Proc. Natl. Acad. Sci. USA 93, No. 26, 15004--15008 (1996; Zbl 1125.11346).

MSC 2000:: *11E25 Sums of squares, etc
33E05 Elliptic functions and integrals
05A15 Combinatorial enumeration problems
33D70 Basic hypergeometric functions and integrals in several variables
11B65 Binomial coefficients, etc.
11F27 Theta series; Weil representation
33D67 Basic hypergeometric functions associated with root systems

Citations: Zbl 1125.11316; Zbl 1125.11346

Cited in: Zbl 1218.11040 Zbl 1151.11017 Zbl 1133.11031 Zbl 1125.11316 Zbl 1125.11321 Zbl 1125.11319

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