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,⋅⋅⋅ ).](a150x.png) | (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)](a151x.png) | (1.2) |
where A, B, C are polynomials of degree ≦ 2 and λ is a parameter.
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