Method for Obtaining Coefficients of Powers of Multivariate Generating Functions

Abstract

There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is the application of compositae of generating functions. In previous studies, we have introduced a generalization for the compositae of multivariate generating functions and have defined basic operations on the compositae of bivariate generating functions. The use of these operations helps to obtain explicit formulas for compositae and coefficients of generating functions in two variables. In this paper, we expand these operations on compositae to the case of generating functions in three variables. In addition, we describe a way of applying compositae to obtain coefficients of rational generating functions in several variables. To confirm the effectiveness of using the proposed method, we present detailed examples of its application in obtaining explicit formulas for the coefficients of a generating function related to the Aztec diamond and a generating function related to the permutations with cycles.

An Approach to Multidimensional Discrete Generating Series

“…Some properties of linear difference operators whose coefficients have the form of infinite two-sided sequences over a field of characteristic zero are considered in [7]. An effective method of obtaining explicit formulas for the coefficients of a generating function related to the Aztec diamond and a generating function related to the permutations with cycles was derived in [8,9]. Using the notion of amoeba [10] of the characteristic polynomial of a difference equation, a description for the solution space of a multidimensional difference equation with constant coefficients was obtained in [11].…”