Some identities involving theta functions

JOURNAL OF NUMBER THEORY

卷:129

Article

Alaca, Ayse¹  Alaca, Saban¹  Williams, Kenneth S.¹ 

[1] Carleton Univ, Ctr Res Algebra & Number Theory, Sch Math & Stat, Ottawa, ON K1S 5B6, Canada

关键词: Theta function identities;    Sums of squares;    Eisenstein series;   

DOI  :  10.1016/j.jnt.2008.10.006

来源: Elsevier

【 摘 要 】

Let phi(q) = Sigma(infinity)(n=-infinity) q(n2) (|q| < 1). For k is an element of N it is shown that there exist k rational numbers A(k, 0), ... , A(k, k-1) such that 1 + 4/E-2k Sigma(infinity)(n=1)(Sigma(d is an element of N) (d|n)(-4/d)d(2k))q(n) = Sigma(k-1)(j=0) A(k, j)phi(4j+2)(q)phi(4k-4j)(-q) where E-2k is an Euler number. Similarly it is shown that there exist k + 1 rational numbers B(k, 0), ... , B(k, k) such that Sigma(infinity)(n=1)(Sigma(d is an element of N) (d|n)(-4/n/d)d(2k))q(n) = Sigma(k-1)(j=0) B(k, j)phi(4j+2)(q)phi(4k-4j)(-q) Recurrence relations are given for the A (k, j) and B (k, j). (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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