Method for Obtaining Coefficients of Powers of Bivariate Generating Functions
Abstract
:1. Introduction
- Potential polynomials introduced by Comtet [2]: The potential polynomial is the kth power of an exponential generating function (k is a complex number):For the coefficients of potential polynomials, there is a relationship with the Bell polynomials, but operations on such polynomials are not defined.
- Riordan arrays introduced by Shapiro et al. [20]: A Riordan array is a pair of generating functions where and . It forms an infinite matrix where . If we consider the associated subgroup of the Riordan group , then we get orHowever, there are no universal rules for obtaining explicit formulas for .
- Power matrices introduced by Knuth [21]: The power matrix of a given power series is the infinite array of coefficients . Thus, the kth power of can be presented in the formIn addition, there are formulas for obtaining the coefficients for the composition of power series and for a power series of the form . However, the development of this mathematical apparatus is not presented.
- Compositae introduced by Kruchinin [22,23]: The composita of a generating function is a coefficients function of its kth power:For two generating functions and and their compositae and , we can find the composita of the generating function for the following cases:
- addition of generating functions: ;
- multiplication of generating functions: ;
- composition of generating functions: ;
- reciprocation of generating functions: ;
- compositional inversion of generating functions: .
In this way, it is possible to obtain an explicit formula for the composita for various types of generating functions.
2. Composita of a Multivariate Generating Function
3. Operations on Compositae of Bivariate Generating Functions
3.1. Composition of Bivariate Generating Functions
3.2. Addition of Bivariate Generating Functions
3.3. Multiplication of Bivariate Generating Functions
3.4. Reciprocation of Bivariate Generating Functions
3.5. Compositional Inversion of Bivariate Generating Functions
4. Application of Compositae for Obtaining Coefficients of Bivariate Generating Functions
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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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
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 StyleKruchinin, Dmitry, Vladimir Kruchinin, and Yuriy Shablya. 2021. "Method for Obtaining Coefficients of Powers of Bivariate Generating Functions" Mathematics 9, no. 4: 428. https://doi.org/10.3390/math9040428
APA StyleKruchinin, D., Kruchinin, V., & Shablya, Y. (2021). Method for Obtaining Coefficients of Powers of Bivariate Generating Functions. Mathematics, 9(4), 428. https://doi.org/10.3390/math9040428