On arithmetic nature of special values of the incomplete beta function
Acta Arithmetica
MSC: Primary 11J81; Secondary 11J86
DOI: 10.4064/aa240610-6-10
Published online: 19 February 2025
Abstract
We study the arithmetic nature of special values of the incomplete beta function $B_x(a,b)$, defined by the integral $\int_0^xt^{a-1}(1-t)^{b-1}\,dt$ for $a, b \gt 0$ and $0 \leq x \leq 1$. For $x=1$, one recovers the beta function $B(a,b)=\int_0^1t^{a-1}(1-t)^{b-1}\,dt$, for which Schneider proved that $B(a,b)$ is transcendental for any $a,b \in \mathbb Q \setminus \mathbb Z $ such that $a + b \notin \mathbb Z $. However, possible transcendental nature of special values of the incomplete beta function is a delicate question due to its relation to the Gauss hypergeometric function.
