Algebraic independence results for the values of the theta-constants and some identities

Carsten Elsner, Masanobu Kaneko, Yohei Tachiya

Research output: Contribution to journalArticlepeer-review

Abstract

In the present work, we give algebraic independence results for the values of the classical theta-constants ϑ2(τ), ϑ3(τ), and ϑ4(τ). For example, the two values ϑα(mτ) and ϑβ(nτ) are algebraically independent over Q for any τ in the upper half-plane when eπiτ is an algebraic number, where m, n ≥ 1 are integers and α, β ∈ {2, 3, 4} with (m, α) ≠ (n, β). This algebraic independence result provides new examples of transcendental numbers through some identities found by S. Ramanujan. We additionally give some explicit identities among the three theta-constants in particular cases.

Original languageEnglish
Pages (from-to)71-80
Number of pages10
JournalJournal of the Ramanujan Mathematical Society
Volume35
Issue number1
Publication statusPublished - Mar 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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