Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

In this paper, we study methods for obtaining explicit formulas for the coefficients of generating functions. To solve this problem, we consider the methods that are based on using the powers of generating functions. We propose to generalize the concept of compositae to the case of generating functions in two variables and define basic operations on such compositae: composition, addition, multiplication, reciprocation and compositional inversion. These operations allow obtaining explicit formulas for compositae and coefficients of bivariate generating functions. In addition, we present several examples of applying the obtained results for getting explicit formulas for the coefficients of bivariate generating functions. The introduced mathematical apparatus can be used for solving different problems that are related to the theory of generating functions.

Automated summary

This paper explores methods for deriving explicit formulas for coefficients of generating functions by utilizing powers of generating functions. The concept of compositae is generalized to bivariate generating functions, and basic operations are defined to obtain explicit formulas for compositae and coefficients of bivariate generating functions. The presented mathematical apparatus can be utilized to solve various problems related to the theory of generating functions.