The Difference Calculus and The NEgative Binomial Distribution

In a previous paper we state the dominant term in the third central moment of the maximum likelihood estimator k of the parameter k in the negative binomial probability function where the probability generating function is (p + 1 - pt){sup -k}. A partial sum of the series {Sigma}1/(k + x){sup 3} is involved, where x is a negative binomial random variate. In expectation this sum can only be found numerically using the computer. Here we give a simple definite integral in (0,1) for the generalized case. This means that now we do have a valid expression for {radical}{beta}{sub 11}(k) and {radical}{beta}{sub 11}(p). In addition we use the finite difference operator {Delta}, and E = 1 + {Delta} to set up formulas for low order moments. Other examples of the operators are quoted relating to the orthogonal set of polynomials associated with the negative binomial probability function used as a weight function.