DOI: 10.68381/jca25060 ISSN: 0944-6532

Directional Convexity and Characterizations of Beta and Gamma Functions

Martin Himmel, Janusz Matkowski
Show PDF Cite

The logarithmic convexity of restrictions of the Beta function to rays parallel to the main diagonal and the functional equation

\varphi (x+1) = \frac{x(x+k)}{(2x+k+1)(2x+k)}\, \phi(x),\ \ x>0, φ ( x + 1 ) = x ( x + k ) ( 2 x + k + 1 ) ( 2 x + k )   ϕ ( x ) ,    x > 0 ,
for
k>0 k > 0
allow to get a characterization of the Beta function. This fact and the notion of the beta-type function lead to a new characterization of the Gamma function.

More from our Archive