Research Related To Identities Involving Harmonic Numbers

The research on identities involving generalized harmonic numbers has always been a hot field.In this paper,by applying the properties of Gamma functions,Polygamma functions and Bell polynomials and their relations and making use of the method of differential and integral,we obtain some new identities.In this process,we extend the Gauss’s summation theorem by the method of partial fractional decomposition,so that the extended theorem has a wider range of applications for solving this kind of problem.Secondly,according to the definition of hypergeometric series itself,we obtain a method of evaluating the values of a family of hypergeometric series by disassembling and then combining this hypergeometric series.At the same time,according to the relationship between the derivative of the shifted factorial,the Bell polynomials and the generalized harmonic numbers,we obtain an integral expression of the generalized harmonic numbers.Basing on the identity,we obtain many meaningful identities.The specific contents are as follows:1)By using the expression of partial decomposition of a fraction,we extend the Gauss’s summation theorem,and then obtain the Euler summation closed formulas of(?)Based on this theorem,a closed formulas for the summation of two types of infinite series and are obtained by introducing parameters and then differentiating.2)According to the extended Gauss’s summation theorem and the definition of hypergeometric series,and then applying the method of partial fraction decomposition,the value of a certain type of hypergeometric series is calculated by disassembling it and then combining it.Finally,we deduce the method of calculating the value of(?)3)Based on the relationship between the derivative of the shifted factorial(1+x)n and the generalized harmonic numbers,an integral expression of the generalized harmonic numbers is proved.And then according to the expression,we obtain the method of Log-integral transformation.Many new identities are obtained by making use of the method.