Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya
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Dmitry Kruchinin: Department of Complex Information Security of Computer Systems, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
Vladimir Kruchinin: Institute of Innovation, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
Yuriy Shablya: Department of Complex Information Security of Computer Systems, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia

Mathematics, 2021, vol. 9, issue 4, 1-17

Abstract: 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.

Keywords: formal power series; composition of generation functions; bivariate generating function; composita; explicit formula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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