Special values of the Gamma function at CM points

Abstract

Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z) when \(\mathbb{Q}(z)\) is an imaginary quadratic field. We also study special values of the digamma function, ψ(z), and the polygamma functions, ψt(z). As part of our analysis we will see that certain infinite products $$\prod_{n=1}^{\infty} \frac{A(n)}{n^t} $$ can be evaluated explicitly and are transcendental for \(A(z) \in\overline{\mathbb{Q}}[z]\) with degree t and roots from an imaginary quadratic field. Special cases of these products were studied by Ramanujan. Additionally, we explore the implications that some conjectures of Gel’fond and Schneider have on these values and products.