Opuscula Math.
24
, no. 1
(), 35-41
Opuscula Mathematica

# 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.
Mathematics Subject Classification: 39A10, 33B15, 26A51.