Milne, Stephen C. Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. (English) Zbl 1125.11315 Ramanujan J. 6, No. 1, 7-149 (2002). 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). Cited in 7 ReviewsCited in 49 Documents MSC: 11E25 Sums of squares and representations by other particular quadratic forms 33E05 Elliptic functions and integrals 05A15 Exact enumeration problems, generating functions 33D70 Other basic hypergeometric functions and integrals in several variables 11B65 Binomial coefficients; factorials; \(q\)-identities 11F27 Theta series; Weil representation; theta correspondences 33D67 Basic hypergeometric functions associated with root systems Citations:Zbl 1125.11316; Zbl 1125.11346 PDFBibTeX XMLCite \textit{S. C. Milne}, Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315) Full Text: DOI arXiv Digital Library of Mathematical Functions: §27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory Online Encyclopedia of Integer Sequences: Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4. Number of ways of writing n as a sum of 5 squares. Number of ways of writing n as a sum of 6 squares. Number of ways of writing n as a sum of 8 squares. Number of ways of writing n as a sum of 16 squares. Number of ways of writing n as a sum of 24 squares. Ramanujan’s tau function (or Ramanujan numbers, or tau numbers). Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3). Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed). Number of ways of writing n as a sum of 7 squares. Number of ways of writing n as a sum of 9 squares. From Euler’s Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m). A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.