Some Properties And Applications Of Q Gamma Functions And Second Class BELL Polynomials

In recent years,many famous domestic and foreign scholars have engaged in the research about complete monotonicity relating the Gamma function andψ function,which has become one of the mathematics knowledge of new points.Complete monotonicity and logarithmic complete monotonicity on Gamma func-tion and ψ function,have played vital role in proofing of some important inequal-ities,strengthening and promoting.Other,in combinatorial mathematics,Bell polynomials and the Catalan number meet the large number of special sequence identity,and have a wide range of applications in combinatorial mathematics,number theory,numerical analysis,and other fields.So in this paper,more above knowledge is involved in the contents.This paper mainly divided into five parts,the first part mainly comment on two papers involving the q-trigamma function.The second part mainly studies a gamma function and ψ function q-after complete monotonicity properties for the function.The third part mainly find necessary con-ditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmic completely monotonic.The fourth part establish several explicit formulas for special values of the Bell polynomials of the second kind,connect these formulas with the Bessel polynomials,and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions,such as the since,cosine,exponential,logarithm,and so on.The fifth part is summary and outlook.