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70 Gegenbauer weight w(λ) (z) = (1 − z 2 )λ− 2 , λ > − 12 . Here (cf. 1) √ Γ(λ + 12 ) , π Γ(λ + 1) k(k + 2λ − 1) , bk = 4(k + λ)(k + λ − 1) m0 = k = 1, 2, 3, . . 20) one finds by induction 1 Γ(λ + 12 ) , θ0 = √ π Γ(λ + 1) For λ = 1 2 1 Γ( 12 (k + 2))Γ(λ + 12 (k + 1)) , λ+k Γ( 12 (k + 1))Γ(λ + 12 k) k ≥ 1. e. 3 θk = 2 2k + 1 Γ( 12 (k + 2)) Γ( 12 (k + 1)) 2 , k≥0 (w(z) = 1). Zeros Similarly as for ordinary and Sobolev orthogonal polynomials, the zeros of πn ( · ) = πn ( · ; w), here too, can be characterized as eigenvalues of a certain matrix, this time a complex matrix, giving rise to complex eigenvalues.

Rational Szeg¨ o quadrature formulae exact on spaces of rational functions having prescribed poles are discussed in Bultheel, Gonz´ alez-Vera, Hendriksen, and Nj˚ astad (2001). Formally orthogonal polynomials on arcs in the complex plane and related (complex) Gaussian quadrature formulae are consid- 50 BASIC THEORY ered in Saylor and Smolarski (2001) in connection with the biconjugate gradient algorithm of numerical linear algebra. Orthogonal polynomials of several variables and matrix orthogonal polynomials are beyond the scope of this book.

1). ) Once distinct nodes have been found that satisfy these constraints, the respective weights λν , by (a), can be found by interpolation. 45 is optimal. Indeed, k = n + 1, according to (b), would require orthogonality of ωn to all polynomials of degree ≤ n, in particular orthogonality onto itself. This is impossible. 1) with k = n, that is, having degree of exactness d = 2n − 1, is called the Gauss quadrature rule with respect to the measure dλ. Condition (b) then shows that ωn (t) = πn (t; dλ), that is, the nodes τν are the zeros of the polynomial of degree n orthogonal with respect to the measure dλ.

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