2001/12/13 by A. Raouf Chouikha, Chouikha, A. Raouf
Mathematics · #11F11 #11F20 #34A20 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary: 33E05 #Secondary: 33E20 #math.CA #math.NT #msc:11F11 #msc:11F20 #msc:33E05 #msc:33E20 #msc:34A20
paper · pdf · doi:10.48550/arxiv.math/0112137
23 pages
openalex publication_date 2001/12/13 · arxiv created 2006/05/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the classical theta function θ4 may be expressed as θ4(v,τ) = θ4(0,τ) exp[- ∑p≥ 1 ∑k≥ 0 \frac 1p (\frac sin πv(sin (k+1/2)πτ))2p]. We obtain an analogous expansion for the three other theta functions since they are related. These results have several consequences. In particular, an expansion of the Weierstrass elliptic function will be derived. Actions of the modular group and other arithmetical properties will also be considered. Finally using a new expression for the Rogers-Ramanujan continued fraction we produce a simple proof of a Rogers identity. \it Key words and phrases : theta functions, elliptic functions, q-series, Fourier series, continued fractions