Novel Formulas of Schröder Polynomials and Their Related Numbers

Waleed Mohamed Abd-Elhameed () and Amr Kamel Amin
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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

Mathematics, 2023, vol. 11, issue 2, 1-23

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.

Keywords: Schröder numbers; Schröder polynomials; orthogonal polynomials; connection and linearization coefficients; generalized hypergeometric functions; symbolic computation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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