TY  - JOUR
ID  - Choczewski2004
LB  - Choczewski2004
AU  - Choczewski, Bodgan
AU  - Wach-Michalik, Anna
TI  - Euler's Beta function diagonalized and a related functional equation
ST  - Euler's Beta function diagonalized and a related functional equation
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2004
VL  - 24
IS  - 1
SP  - 35
EP  - 41
DO  - http://dx.doi.org/10.7494/OpMath.2004.24.1.35
UR  - http://dx.doi.org/10.7494/OpMath.2004.24.1.35
AB  - 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 \(\varphi =\beta\).
KW  - Euler's Beta function
KW  - diagonalization
KW  - functional equations
KW  - convex functions
SN  - 1232-9274
ER  -