IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i4p428-d503404.html
   My bibliography  Save this article

Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/4/428/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/4/428/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Francesco Aldo Costabile & Maria Italia Gualtieri & Anna Napoli, 2022. "Polynomial Sequences and Their Applications," Mathematics, MDPI, vol. 10(24), pages 1-3, December.
    2. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kucukoglu, Irem & Simsek, Buket & Simsek, Yilmaz, 2019. "Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature," Applied Mathematics and Computation, Elsevier, vol. 344, pages 150-162.
    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. AERTS, Babiche & CORNELISSENS, Trijntje & SÖRENSEN, Kenneth, 2020. "Solving the joint order batching and picker routing problem, as a clustered vehicle routing problem," Working Papers 2020003, University of Antwerp, Faculty of Business and Economics.
    4. Diefenbach, Heiko & Grosse, Eric H. & Glock, Christoph H., 2024. "Human-and-cost-centric storage assignment optimization in picker-to-parts warehouses," European Journal of Operational Research, Elsevier, vol. 315(3), pages 1049-1068.
    5. Rajabighamchi, Farzaneh & van Hoesel, Stan & Defryn, Christof, 2023. "Graph reduction for the planar Travelling Salesman Problem," Research Memorandum 004, Maastricht University, Graduate School of Business and Economics (GSBE).
    6. Briant, Olivier & Cambazard, Hadrien & Cattaruzza, Diego & Catusse, Nicolas & Ladier, Anne-Laure & Ogier, Maxime, 2020. "An efficient and general approach for the joint order batching and picker routing problem," European Journal of Operational Research, Elsevier, vol. 285(2), pages 497-512.
    7. Bock, Stefan, 2024. "Vehicle routing for connected service areas - a versatile approach covering single, hierarchical, and bi-criteria objectives," European Journal of Operational Research, Elsevier, vol. 313(3), pages 905-925.
    8. Yuriy Shablya & Dmitry Kruchinin & Vladimir Kruchinin, 2020. "Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application," Mathematics, MDPI, vol. 8(6), pages 1-21, June.
    9. Mustapha Haouassi & Yannick Kergosien & Jorge E. Mendoza & Louis-Martin Rousseau, 2022. "The integrated orderline batching, batch scheduling, and picker routing problem with multiple pickers: the benefits of splitting customer orders," Flexible Services and Manufacturing Journal, Springer, vol. 34(3), pages 614-645, September.
    10. Maximilian Schiffer & Nils Boysen & Patrick S. Klein & Gilbert Laporte & Marco Pavone, 2022. "Optimal Picking Policies in E-Commerce Warehouses," Management Science, INFORMS, vol. 68(10), pages 7497-7517, October.
    11. Neves-Moreira, Fábio & Amorim, Pedro, 2024. "Learning efficient in-store picking strategies to reduce customer encounters in omnichannel retail," International Journal of Production Economics, Elsevier, vol. 267(C).
    12. Rajabighamchi, Farzaneh & van Hoesel, Stan & Defryn, Christof, 2023. "The order picking problem under a scattered storage policy," Research Memorandum 006, Maastricht University, Graduate School of Business and Economics (GSBE).
    13. Saylam, Serhat & Çelik, Melih & Süral, Haldun, 2024. "Arc routing based compact formulations for picker routing in single and two block parallel aisle warehouses," European Journal of Operational Research, Elsevier, vol. 313(1), pages 225-240.
    14. Boysen, Nils & de Koster, René & Füßler, David, 2021. "The forgotten sons: Warehousing systems for brick-and-mortar retail chains," European Journal of Operational Research, Elsevier, vol. 288(2), pages 361-381.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:428-:d:503404. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.