Lattice Sums

Glasser, M. L.; Zucker, I. J.

doi:10.1016/b978-0-12-681905-2.50008-6

Cited by 91 publications

(81 citation statements)

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“…Finally, s λ and s ν are the Schur functions in (6.12), with the partitions λ and ν given by 90) and,…”

Section: The Schur Function Form Of Sums Of Squares Identitiesmentioning

confidence: 99%

“…The review article [222] presents many questions connected with representations of integers as sums of squares. Direct applications of sums of squares to lattice point problems and crystallography models in theoretical chemistry can be found in [28,90]. One such example is the computation of the constant Z N that occurs in the evaluation of a certain Epstein zeta function, needed in the study of the stability of rare gas crystals, and in that of the so-called Madelung constants of ionic salts.…”

Section: Introductionmentioning

confidence: 99%

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

Milne

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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“…Finally, s λ and s ν are the Schur functions in (6.12), with the partitions λ and ν given by 90) and,…”

Section: The Schur Function Form Of Sums Of Squares Identitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Milne

2002

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

View full text Add to dashboard Cite

show abstract

“…In the case under consideration the Epstein zeta function has a simple pole at ν = . Note that, from the functional equation for the Epstein function [17], one can check easily that C (d|ν), defined in (4), obeys the symmetry property (3).…”

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confidence: 99%

“…On the other hand the sum in the right hand side of equation (4) can be expressed in terms of the Epstein zeta function [17] …”

mentioning

confidence: 99%