The Summation of Series of Hyperbolic Functions

Zucker, I. J.

doi:10.1137/0510019

“…In these two cases at least, we see that the problem reduces to the calculation of the sums, (7) and (8), which are the subjects of the papers by Kiyek and Schmidt [5], Ling [6] and Zucker, [7]. Values of the sum, Q g , for specific g are quite old.…”

Section: Resumé Of Earlier Workmentioning

confidence: 99%

“…An alternative way of arranging the evaluation of the Q g is to rewrite the definition (7) as a q-series as done by Zucker, [7]. We re-express his approach somewhat to fit our requirements.…”

Section: Hyperbolic Summationsmentioning

confidence: 99%

“…Zucker, [7], from results of Ramanujan, says that E, as well as K, can be expressed in terms of algebraic numbers and Gamma functions at singular moduli.…”

Section: Temperature Inversionmentioning

confidence: 99%

“…cf Zucker, [7], who uses the expansion of nc 2 to extract the Lambert series. As has been mentioned, our previous analysis [1] expanded the degeneracies in powers of n 2 , n being the mode label, and treated each power separately making use of classic elliptic facts.…”

Section: Hyperbolic Summationsmentioning

confidence: 99%

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Kirsten

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2008

Journal of Mathematical Physics

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Our earlier results on the temperature inversion properties and the ellipticisation of the finite temperature internal energy on odd spheres are extended to orbifold factors of odd spheres and then to other thermodynamic quantities, in particular to the specific heat. The behaviour under modular transformations is facilitated by the introduction of a modular covariant derivative and it is shown that the specific heat on any odd sphere can be expressed in terms of just three functions. It is also shown that the free energy on the circle can be written elliptically.

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“…This background material is contained in [2,15,16,21,23,43,63,79,92,93,103,104,111,119,125,134,149,153,184,203,206,214,247,249,258]. In order to make the connection with divisor sums, we follow Jacobi in emphasizing Lambert series ∞ r=1 f (r)q r /(1 − q r ), as defined in [8,Ex.…”

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

show abstract

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

Milne¹

2002

Developments in Mathematics

View full text Add to dashboard Cite

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

The Summation of Series of Hyperbolic Functions

Cited by 44 publications

References 4 publications

Elliptic aspects of statistical mechanics on spheres

Elliptic aspects of statistical mechanics on spheres

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

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