CHARACTERIZATION OF POLYNOMIALS VIA A RAISING OPERATOR

Abstract

This paper investigates a first-order linear differential operator 𝒥𝜉, where 𝜉 = (𝜉1, 𝜉2)\in (C^2\(0,0), and 𝐷 := 𝑑/𝑑𝑥. The operator is defined as 𝒥𝜉 := 𝑥(𝑥𝐷+ I) + 𝜉1 I + 𝜉2𝐷, with I representing the identity on the space of polynomials with complex coefficients. The focus is on exploring the 𝒥𝜉-classical orthogonal polynomials and analyzing properties of the resulting sequences. This work contributes to the understanding of these polynomials and their characteristics.

Keywords

• orthogonal polynomials
• сlassical polynomials
• second- order differential equation
• raising operator