Some identities involving theta functions
type: Item
instance of: scholarly article
title: Some identities involving theta functions
zbMATH Open document ID: 1173.11027
author: Ayşe Alaca
author: Şaban Alaca
author: Kenneth S. Williams
published in: Journal of Number Theory
publication date: 2009-05-28T00:00:00Z
review text: Let \(q\) be a complex variable with \(|q|<1\). Let \(\varphi(q):=\sum_{n=-\infty}^{\infty}q^{n^2}\) be the theta function, and for \(k\geq 1\) let \[ F_k(q):=1+\frac{4}{E_{2k}}\sum_{n=1}^\infty\left(\sum_{d|n}\left(\frac{-4}{d}\right)d^{2k}\right)q^n, \] where \(E_{2k}\) is the Euler number and \(\left(\frac{-4}{d}\right)\) is the Legendre-Jacobi-Kronecker symbol, \[ G_k(q):=\sum_{n=1}^\infty \left(\sum_{d|n}\left(\frac{-4}{n/d}\right)d^{2k}\right)q^n. \] The authors prove that \(F_k(q)\) and \(G_k(q)\) can be expressed as a polynomial in \(\varphi(q)\) and \(\varphi(-q)\) with rational coefficients and give recurrence relations for these coefficients.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 5559088
zbMATH Keywords: theta function identities
zbMATH Keywords: sums of squares
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.10.006
OpenAlex ID: W2119502632
cites work: Q3504975
cites work: Q3218192
cites work: Q5485643
cites work: Q3693570
cites work: Q5804260
cites work: Q5164805
cites work: Elementary proofs of relations between Eisenstein series
cites work: A quick combinatorial proof of Eisenstein series identities
DOI: 10.1016/J.JNT.2008.10.006
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description: scientific article; zbMATH DE number 5559088