Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions

Abstract

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 τ (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 η-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 ℓ nonterminating 6 φ 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

Representations of integers as sums of an even number of squares

“…In [8], [9], using similar methods, he also proved a conjecture of Kac and Wakimoto on the number of representations of n as a sum of triangular numbers. This conjecture was also proved independently in a different way, using the theory of modular forms, by Zagier [14].…”

“…Recently, Milne [7][8][9] took up the subject again from a completely different point of view and using tools from the theory of elliptic functions, Lie theory, the theory of hypergeometric functions and other devices obtained exact formulas for r s (n) whenever s = 4j 2 or s = 4j 2 + 4j with j ∈ N, in terms of explicit finite sums of products of j modified elementary divisor functions. In [8], [9], using similar methods, he also proved a conjecture of Kac and Wakimoto on the number of representations of n as a sum of triangular numbers.…”

“…There are also corresponding conjectures for s ≡ 2, 4, 6 (mod 8). The paper of Chan and Chua was motivated by the new 2 × 2 determinant formula for θ 24 in Theorem 1.6 of [9] and the n = 2 case of Theorem 2.3 in [8]. In [3] the case s = 32 is treated in detail.…”

A nonlinear oscillatory system subjected to driving forces of elliptic type

“…Similar notation is used for the elliptical functions sn and dn, namely, sn j, f,F and dn j, f,F . Substituting Equations (2) and (3) into Equation (1) and recalling the relations [9,10] …”

“…Similarly, we compute the average of Equation (9) with respect to 4K 1 and then with respect to 4K F and finally, we compute the average of Equation (10) with respect to 4K 2 and then with respect to 4K f . These computations allow us to reduce Equation (9) and Equation (10) to the following expressions:…”