*(English)*Zbl 1125.11315

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 $4{n}^{2}$ 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 $4n(n+1)$, $2{n}^{2}+n$, $2{n}^{2}-n$ 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 $4{n}^{2}$ 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:

11E25 | Sums of squares, etc |

33E05 | Elliptic functions and integrals |

05A15 | Exact enumeration problems, generating functions |

33D70 | Basic hypergeometric functions and integrals in several variables |

11B65 | Binomial coefficients, etc. |

11F27 | Theta series; Weil representation; theta correspondences |

33D67 | Basic hypergeometric functions associated with root systems |