Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant θ₃

2019/04/09 by Robert F. Scherer, Scherer, Robert
Mathematics · #11B83 #11F37 #14K25 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1904.04509

openalex publication_date 2019/04/09 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ₃, around the point x = 1. He discovered a new integer sequence, (d(n))₀^∞=1, 1, -1, 51, 849, -26199, …, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)≡ (-1)ⁿ⁺¹(mod 5) for all n≥ 1,and that for any prime p≡ 3 (mod 4), d(n) vanishes modulo p for all large enough n.

Related

• The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function θ₃
• The Taylor coefficients of the Jacobi theta constant θ₃
• Romik's Conjecture for the Jacobi Theta Function
• Taylor coefficients of the Jacobi θ₃( q ) function
• Congruences for Domb and Almkvist-Zudilin numbers