Formulas for characteristic function and moment generating functions of beta type distribution

We study on the beta type distribution associated with the Bernstein type basis functions and the beta function, which was defined by authors (Yalcin and Simsek in Symmetry 12(5):779, 2020). The aim of this paper is to define characteristic function of the Beta type distribution. Using interesting integral formulas, we also give many new formulas and relations for this characteristic function. Furthermore, by using moment generating function and characteristic functions, we not also present Kurtosis Excess for beta type distribution, but also give some new identities for the moment of the Beta type distribution. Finally, we give relations among expected values for the logarithm of random variable, the Stirling numbers, the Catalan numbers, the digamma function, the beta function, and the gamma function. We also give remarks and comments on the special values of our new formulas and relations.

Automated summary

This paper studies the beta type distribution associated with the Bernstein type basis functions and the beta function, and defines its characteristic function. Using integral formulas, new formulas and relations for the characteristic function are given, as well as the kurtosis excess and new identities for the moments of the beta type distribution. Relations among expected values for the logarithm of random variables, Stirling numbers, Catalan numbers, digamma function, beta function, and gamma function are also discussed.