# Journal of the Ramanujan Mathematical Society

Volume 35, Issue 1, March 2020 pp. 71–80.

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

**Authors**:
Carsten Elsner, Masanobu Kaneko and Yohei Tachiya

**Author institution:**Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, D-30173 Hannover, Germany

**Summary: **
In the present work, we give algebraic independence results for
the values of the classical theta-constants ϑ 2 (τ),
ϑ 3(τ), and ϑ 4(τ). For example, the two
values ϑ α (mτ) and ϑ β (nτ) are
algebraically independent over Q for any τ in the
upper half-plane when e{π iτ} 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.

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