Quadratic forms and a product-to-sum formula

• Autorzy: Kenneth S. Williams
• Języki publikacji: EN

Abstrakty

EN

Let q ∈ ℂ satisfy |q| < 1. If $f(q) = ∑_{n=0}^{∞} f_n q^n$ we write $[f(q)]_n = f_n$. We prove a general product-to-sum formula which includes known formulae such as
$[q∏_{k=1}^{∞} (1-q^{2k})^3 (1-q^{6k})^3 _n = ∑_{{(x_1,x_2)∈ ℤ^2 \atop x_1^2+3x_2^2=n}} 1/2(x_1^2-3x_2^2)$
and
$[q∏_{k=1}^{∞}(1-q^{4k})^6]_n = ∑_{{(x_1,x_2)∈ ℤ^2 \atop x_1^2+4x_2^2=n}} 1/2(x_1^2-4x_2^2).

Kategorie tematyczne

• 11F25: Hecke-Petersson operators, differential operators (one variable)
• 11F20: Dedekind eta function, Dedekind sums
• 11E25: Sums of squares and representations by other particular quadratic forms

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 158
• Numer: 1
• Strony: 79-97

Daty

wydano

2013

Twórcy

autor

Kenneth S. Williams

• Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

Identyfikatory

DOI

10.4064/aa158-1-5