Novel Formulas of Schroder Polynomials and Their Related Numbers

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