The Taylor coefficients of the Jacobi theta constant θ₃

2018/07/16 by Dan Romik, Romik, Dan
Mathematics · #11F37 #14K25 #30B10 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F37 #msc:14K25 #msc:30B10

paper · pdf · doi:10.48550/arxiv.1807.06130

19 pages. V2 update: journal version, corrected several typos from original version

openalex publication_date 2018/07/16 · openalex created_date 2018/08/03 · arxiv created 2018/10/17 · arxiv updated 2018/10/19 · openalex updated_date 2026/07/28

Abstract

We study the Taylor expansion around the point x=1 of a classical modular form, the Jacobi theta constant θ₃. This leads naturally to a new sequence (d(n))_n=0^∞=1,1,-1,51,849,-26199,… of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of θ₃. We prove several results about the numbers d(n) and conjecture that they satisfy the congruence d(n)≡ (-1)^n-1 (\textrmmod 5) and other similar congruence relations.

Related

• Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant θ₃
• Taylor coefficients of the Jacobi θ₃( q ) function
• The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function θ₃
• Romik's Conjecture for the Jacobi Theta Function
• Congruence properties of Taylor coefficients of modular forms