On the Representation of a Number as a Sum of Squares

Mathews, G. B.

doi:10.1112/plms/s1-27.1.55

“…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%

“…Similarly, explicit formulas for r s (n) have been found for (odd) s < 32. Alternate, elementary approaches to sums of squares formulas can be found in [154,211,[226][227][228][229].…”

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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“…As an application of Theorem 5, we recover the following classical result of Glaisher [4] and Mathews [9].…”

Section: Corollary 1 Suppose S Is a Positive Integer Then For N 1 mentioning

confidence: 89%

Divisors of modular forms on Γ0(4)

Atkinson

¹

2005

Journal of Number Theory

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There is a relationship between the values of a sequence of modular functions at points in the divisor of a meromorphic modular form and the exponents of its infinite product expansion. We make this relationship explicit for the case of modular forms on the congruence subgroup 0 (4). We also consider some applications to classical number theoretic functions such as the number of representations of an integer as a sum of squares.

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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

show abstract

On the Representation of a Number as a Sum of Squares

Cited by 4 publications

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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

Divisors of modular forms on Γ0(4)

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

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