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Novel Formulas of Schröder Polynomials and Their Related Numbers

Author

Listed:
  • Waleed Mohamed Abd-Elhameed

    (Department of Mathematics, College of Science, University of Jeddah, Jeddah 21589, Saudi Arabia)

  • Amr Kamel Amin

    (Department of Basic Sciences, Adham University College, Umm AL-Qura University, Makkah 28653, Saudi Arabia
    Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt)

Abstract

This paper explores the Schröder polynomials, a class of polynomials that produce the famous Schröder numbers when x = 1 . The three-term recurrence relation and the inversion formula of these polynomials are a couple of the fundamental Schröder polynomial characteristics that are given. The derivatives of the moments of Schröder polynomials are given. From this formula, the moments of these polynomials and also their high-order derivatives are deduced as two significant special cases. The derivatives of Schröder polynomials are further expressed in new forms using other polynomials. Connection formulas between Schröder polynomials and a few other polynomials are provided as a direct result of these formulas. Furthermore, new expressions that link some celebrated numbers with Schröder numbers are also given. The formula for the repeated integrals of these polynomials is derived in terms of Schröder polynomials. Furthermore, some linearization formulas involving Schröder polynomials are established.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed & Amr Kamel Amin, 2023. "Novel Formulas of Schröder Polynomials and Their Related Numbers," Mathematics, MDPI, vol. 11(2), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:468-:d:1036886
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    References listed on IDEAS

    as
    1. Sedaghat, S. & Mashayekhi, S., 2022. "Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    2. Waleed Mohamed Abd-Elhameed, 2022. "Novel Formulae of Certain Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 10(22), pages 1-25, November.
    3. Waleed Mohamed Abd-Elhameed & Badah Mohamed Badah, 2021. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas," Mathematics, MDPI, vol. 9(13), pages 1-28, July.
    4. Feng Qi & Xiao-Ting Shi & Bai-Ni Guo, 2018. "Integral Representations of the Large and Little Schröder Numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(1), pages 23-38, March.
    5. W. M. Abd-Elhameed & N. A. Zeyada, 2022. "New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 1006-1016, December.
    6. Waleed Mohamed Abd-Elhameed & Andreas N. Philippou & Nasr Anwer Zeyada, 2022. "Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals," Mathematics, MDPI, vol. 10(13), pages 1-18, July.
    7. Francesco Aldo Costabile & Maria Italia Gualtieri & Anna Napoli, 2021. "General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints," Mathematics, MDPI, vol. 9(9), pages 1-29, April.
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