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Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

Author

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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)

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.

Suggested Citation

  • Dmitry Kruchinin & Vladimir Kruchinin & Yuriy Shablya, 2021. "Method for Obtaining Coefficients of Powers of Bivariate Generating Functions," Mathematics, MDPI, vol. 9(4), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:428-:d:503404
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    References listed on IDEAS

    as
    1. Yilmaz Simsek & Mehmet Acikgoz, 2010. "A New Generating Function of ( ) Bernstein-Type Polynomials and Their Interpolation Function," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-12, March.
    2. Dmitry V. Kruchinin & Yuriy V. Shablya, 2015. "Explicit Formulas for Meixner Polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2015, pages 1-5, October.
    3. Cambazard, Hadrien & Catusse, Nicolas, 2018. "Fixed-parameter algorithms for rectilinear Steiner tree and rectilinear traveling salesman problem in the plane," European Journal of Operational Research, Elsevier, vol. 270(2), pages 419-429.
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    Cited by:

    1. Dmitry Kruchinin & Vladimir Kruchinin & Yuriy Shablya, 2023. "Method for Obtaining Coefficients of Powers of Multivariate Generating Functions," Mathematics, MDPI, vol. 11(13), pages 1-15, June.
    2. Francesco Aldo Costabile & Maria Italia Gualtieri & Anna Napoli, 2022. "Polynomial Sequences and Their Applications," Mathematics, MDPI, vol. 10(24), pages 1-3, December.

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