On the representation of a number as a sum of squares

Gundlach, Karl-Bernhard

doi:10.1017/s0017089500003608

“…241-244]. For 2s > 8 modular function methods such as those in [95,99,100,147,172,201], or the more classical elliptic function approach of [14,25,33,34,39,126,127], [155, pp. 140-143], [241, pp.…”

Section: Introductionmentioning

confidence: 99%

“…We have in [167] 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 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews [154], Glaisher [85][86][87][88], Sierpinski [211], Uspensky [226][227][228], Bulygin [33,34], Ramanujan [193], Mordell [172,173], Hardy [99,100], Bell [14], Estermann [64], Rankin [200,201], Lomadze [147], Walton [248], Walfisz [246], Ananda-Rau [3], van der Pol [186], Krätzel [126,127], Bhaskaran [25], Gundlach [95], Kac and Wakimoto [120], and Liu [146].…”

Section: Introductionmentioning

confidence: 99%

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

Milne

¹

2002

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

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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

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“…This problem has a long and interesting history, which is described in the two books [46], [66]. Relevant references include [9], [12], [13], [15], [16], [22 [45], [47]- [52], [60], [61], [63]- [65], [67], [74]- [77], [79], [80], [82], [85].…”

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confidence: 99%

Extensions of the Ramanujan-Mordell formula

Lemire¹

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The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. L'auteur a accorde une licence non exclusive permettant a la Bibliotheque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par I'lnternet, preter, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant.

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

Milne¹

2002

Developments in Mathematics

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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

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On the representation of a number as a sum of squares

Cited by 6 publications

References 12 publications

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

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

Extensions of the Ramanujan-Mordell formula

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

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