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

Sumudu Transform of Dixon Elliptic Functions With Non-Zero Modulus as Quasi C Fractions and Its Hankel Determinants

“…be a sequence in C. Then the following m × m matrices are defined [14,20,25], whose determinants are denoted by respective H (n) m and χ m . The relation between FPS and Regular C fraction is given as lemma [14], (Theorem 7.2, pp 223-226, [20]), [25].…”

“…The relation between FPS and Regular C fraction is given as lemma [14], (Theorem 7.2, pp 223-226, [20]), [25]. Lemma 1.…”

“…Laplace transform of Jacobi elliptic functions expanded as continued fractions and shown their coeffecients are orthogonal polynomials and derived dual Hahn polynomials in [19]. Fourier series and continued fractions expansions of ratis of Jacobi elliptic functions and their Hankel determinants are given in [25] from which different ways of representing sum of square numbers derived in determinant forms in [25]. Laplace transform of bimodular Jacobi elliptic functions expanded as continued fractions in [14] and by modular transformation results were back tracked to unimodular Jacobi elliptic functions in [14].…”

“…The above cubis curve and its relation to Fermat curve is studied for the Urn representation and combinatorics in [15]. Number theory related results followed by [25] for factorial of numbers using DEF given in [4]. DEF relation to trefoil curves and relation to Weierstrass and its derivative functions shown in [23].…”

Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook

“…First, Z.-G. Liu [30] has found a simplified approach to Ramanujan's work in [39]. Secondly, new infinite classes of formulas for r 2k (n) have been discovered by H. H. Chan [20], S. Milne [31], and K. Ono [32].…”