Methodology for using the knowledge base of generating two-variable functions

Mathematical knowledge bases are the development of classical mathematical reference books and encyclopedias, which, in turn, makes them an important tool for conducting various research in mathematical sciences and related fields. At present, there are various knowledge bases of mathematical objects. In this paper, we consider the knowledge base of the generating functions of two variables, which allows us to operate with multivariate objects. The relevance and significance of the work lies in solving various problems related to the mathematical apparatus of generating functions. In the paper we consider the use of the knowledge base of generating functions of two variables for solving problems of operating generating functions and obtaining coefficients for composition, reciprocal and compositional inverse generating functions of two variables and their powers, as well as obtaining explicit expressions for the coefficients of logarithmic derivatives of the generating functions. In addition, an inverse problem is considered aimed at obtaining generating functions for explicit expressions describing their coefficients. The use of the knowledge base of generating two-variable functions contributes to the process of constructing combinatorial generation algorithms for combinatorial objects defined by generating functions of many variables. As an example, the construction of combinatorial generation algorithms for sets defined by the generalized Narayana numbers is shown. The Narayana numbers describe classes of subsets for combinatorial sets defined by the Catalan numbers. In this paper, one of the combinatorial interpretations for the Narayana numbers is chosen - the set of the Dyck paths of length n, which have m peaks.