Vol. 28, No. 2, 1969

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Difference equations for some orthogonal polynomials

H. L. Krall and I. M. Sheffer

Vol. 28 (1969), No. 2, 383–392
Abstract

It is well-known that every orthogonal polynomial set {Pn(x)} satisfies a 3-term recurrence relation of the form

Pn+1(x) = (anx+ bn)Pn(x)+ cnPn−1(x) (n = 1,2,⋅⋅⋅ ).
(1.1)

Some orthogonal sets (polynomials of Jacobi, Hermite and so on) are solutions of differential equations. It will be shown that there exist orthogonal polynomial sets that satisfy 3-term difference equations of the form

A(x)y(x + α)+ B (x )y(x − α )+ C(x)y(x) = λy(x)
(1.2)

where A, B, C are polynomials of degree 2 and λ is a parameter.

Mathematical Subject Classification
Primary: 33.40
Secondary: 39.00
Milestones
Received: 12 March 1968
Published: 1 February 1969
Authors
H. L. Krall
I. M. Sheffer