Novel Formulas of Schröder Polynomials and Their Related Numbers

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.

New Formulas Involving Fibonacci and Certain Orthogonal Polynomials

“…k (x), in powers of x are displayed in Table 1. To obtain the connection formula involving Fibonacci polynomials, the power form representation of the ultraspherical polynomials [19] is utilized along with the inversion formula for Fibonacci polynomials (3) to get…”

“…Over the past few decades, hypergeometric functions have become crucial for solving problems in modern analysis, particularly when special functions are involved. As will be seen here, an important topic in applied analysis is the solution of connection and linear problems involving special polynomials, where determining coefficients that are frequently expressed in terms of hypergeometric functions with specific indices and/or arguments is ultimately required [18,19].…”