The orthogonal polynomials generated by [ceteris omissis]

Abstract

Starting from the generating function, a differential-recurrence relation is derived, which is then combined with the three-term pure recurrence formula (a necessary and sufficient condition for orthogonal polynomials) to obtain a differential equation. It is known that a differential equation of this form can have at most the classical orthogonal polynomials as solutions. The Laguerre and Bessel polynomials do not satisfy this differential equation. However the Hermite polynomials and a special case (α = β) of the Jacobi polynomials i.e., the ultraspherical polynomials do satisfy it.

Keywords

• orthogonal polynomials
• recurrence formula
• generating function
• differential equation
• hermite polynomials
• ultraspher