Directional convexity and characterizations of Beta and Gamma functions

Martin Himmel; Janu sz Matkowski

Directional convexity and characterizations of Beta and Gamma functions

Classical Analysis and ODEs 2015-05-07 v1

Authors: Martin Himmel , Janu sz Matkowski

Abstract

The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation $\phi\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }\phi\left( x\right) ,\ \ \ \ \ \ x>0,$ for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function.

Keywords

directional convexity Beta function Gamma function logarithmic convexity functional equation beta-type function characterization

Cite

@article{arxiv.1505.01472,
  title  = {Directional convexity and characterizations of Beta and Gamma functions},
  author = {Martin Himmel and Janu sz Matkowski},
  journal= {arXiv preprint arXiv:1505.01472},
  year   = {2015}
}

Directional convexity and characterizations of Beta and Gamma functions

Abstract

Keywords

Cite

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