Abstract
In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.
Highlights
IntroductionP ∈ [ X ] , we denote by u, P , the action of a linear functional u in the algebraic dual of [ X ]
Let [ X ] be the vector space of polynomials with complex coefficients
P ∈ [ X ], we denote by u, P, the action of a linear functional u in the algebraic dual of [ X ]
Summary
P ∈ [ X ] , we denote by u, P , the action of a linear functional u in the algebraic dual of [ X ]. [1]): 1) deg Pn = n , 2) the leading coefficient of Pn is equal to 1, 3) u, P= n Pm rnδn,m , n, m ≥ 0, rn ≠ 0, n ≥ 0, where for all polynôme P, deg P denotes its degree. Under these conditions, we say that u is regular. A sequence of monic orthogonal polynomials satisfies a three-term recurrence relation (see [1]): P0 = 1; P1 = X − α1; ( ) Pn+1 = X − αn+1 Pn − ωn Pn−1, n ≥ 1,
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