STIRLING NUMBERS AND THE PARTITION METHOD FOR A POWER SERIES EXPANSION.

The partition method for a power series expansion is a method that utilizes standard integer partitions to evaluate the coefficients in power series expansions and generating functions. Here it is shown how an existing code based on the method can be adapted to deal with partitions homologous to standard integer partitions. Consequently, with the aid of further processing in Mathematica this work presents polynomial expressions for both kinds of Stirling numbers in two cases where: (1) the secondary variable is fixed and (2) it becomes a variable. Interestingly, the second case requires the results produced in the first case for the Stirling numbers of the first kind. In the second case the highest power in the primary variable is found to be dependent upon the secondary variable and the coefficients become polynomials in terms of this variable, whereas in the first case the coefficients are rational. The results represent a major advance on already published results of both kinds of the Stirling numbers due to the introduction of partitions into the analysis. Finally, new results for the related Worpitzky numbers and Stirling polynomials are also presented. [ABSTRACT FROM AUTHOR]