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Higher order coherent pairs. (English) Zbl 1262.42009

The present work concerns a study of (1,1)-coherent pairs of order r.

Let {P n } n0 and {R n } n0 be two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals 𝒰, 𝒱, respectively, satisfy the relation

P n [r] (x)+a n-1,r P n-1 [r] (x)=R n-r (x)+b n-1,r R n-r-1 (x),a n-1,r 0,nr+1,

where f [r] (x)=(n-r)! n!f (r) (x), f (r) (x) denoting the r-th derivative of f(x) for a monic polynomial f of degree n and r.

In this case, the pair (𝒰,𝒱) is called (1,1)-coherent pair of order r; in the particular case when b n,r =0,nn+1, (𝒰,𝒱) is said to be a (1,0)-coherent pair of order r.

When (𝒰, 𝒱) are positive definite linear functionals, there exist two positive Borel measures μ 0 and μ 1 supported on the real line such that

𝒰,f= f(x)dμ 0 ,𝒱,f= f(x)dμ 1 ,

for all polynomials f.

In this case, the authors give necessary and sufficient conditions ensuring that the pair (𝒰,𝒱) is a (1,1)-coherent pair of order r. Moreover, they compute the Sobolev orthogonal polynomials with respect to the inner product

p(x),q(x) λ,r = - + p(x)q(x)dμ 0 +λ - + p (r) (x)q (r) (x)dμ 0 ,λ>0,

where p,q are two polynomials.

For (1,0) (or (1,1))-coherent pairs of order r (𝒰,𝒱), the authors give some algebraic equations connecting the forms 𝒰, 𝒱 and D k 𝒱, where D k 𝒱 denotes the derivatives of the linear functional 𝒱, for k=1,2,3 and rk.

In the case where 𝒰 is classical, they show that 𝒱 is semiclassical of class at most two.

Finally, some relations for the formal Stieltjes series associated with the linear functionals in a (1,1)-coherent pair of order r are given.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
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