Euler's Beta function diagonalized and a related functional equation

Abstract

Euler's Gamma function is the unique logarithmically convex solution of the functional equation \[\varphi(x+1)=x\varphi(x),\quad x\in\mathbb{R}_{+};\quad \varphi(1)=1,\] cf. the Proposition. In this paper we deal with the function \(\beta :\mathbb{R}_{+}\to\mathbb{R}_{+}\), \(\beta (x):=B(x,x)\), where \(B(x,y)\) is the Euler Beta function. We prove that, whenever a function \(h\) is asymptotically comparable at the origin with the function \(a\log +b\), \(a\gt 0\), if \(\varphi :\mathbb{R}_{+}\to\mathbb{R}_{+}\) satisfies equation \[\varphi(x+1)=\frac{x}{2(2x+1)}\varphi(x),\quad x\in\mathbb{R}_{+}\] and the function \(h\circ \varphi\) is continuous and ultimately convex, then \(\varphi =\beta\).

Keywords

• Euler's Beta function
• diagonalization
• functional equations
• convex functions