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Open AccessArticle

Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

1
Department of Complex Information Security of Computer Systems, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
2
Institute of Innovation, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
*
Author to whom correspondence should be addressed.
Academic Editors: Francesco Aldo Costabile, Maria I. Gualtieri and Anna Napoli
Mathematics 2021, 9(4), 428; https://doi.org/10.3390/math9040428
Received: 31 December 2020 / Revised: 10 February 2021 / Accepted: 14 February 2021 / Published: 22 February 2021
(This article belongs to the Special Issue Polynomial Sequences and Their Applications)
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. View Full-Text
Keywords: formal power series; composition of generation functions; bivariate generating function; composita; explicit formula formal power series; composition of generation functions; bivariate generating function; composita; explicit formula
MDPI and ACS Style

Kruchinin, D.; Kruchinin, V.; Shablya, Y. Method for Obtaining Coefficients of Powers of Bivariate Generating Functions. Mathematics 2021, 9, 428. https://doi.org/10.3390/math9040428

AMA Style

Kruchinin D, Kruchinin V, Shablya Y. Method for Obtaining Coefficients of Powers of Bivariate Generating Functions. Mathematics. 2021; 9(4):428. https://doi.org/10.3390/math9040428

Chicago/Turabian Style

Kruchinin, Dmitry; Kruchinin, Vladimir; Shablya, Yuriy. 2021. "Method for Obtaining Coefficients of Powers of Bivariate Generating Functions" Mathematics 9, no. 4: 428. https://doi.org/10.3390/math9040428

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