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Companion linear functionals and Sobolev inner products: a case study. (English) Zbl 1087.42020

Let be the space of all polynomials with real coefficients. A linear functional U: is said to be semiclassical if there exists two polynomials φ and ψ such that the following distributional Pearson equation holds

D(φU)=ψU(1)

where linear functionals ψU and D(φU) are defined as

ψU,p:=U,ψp,D(φU),p=-U,φp ' ,p·

A semiclassical linear functional U is said to be of class s if

min Ω max(degφ-2,degψ-1)=s

where Ω=Ω(U) is the set of all possible pairs of polynomials φ,ψ from Pearson equation (1). Let V,U be two quasi-definite linear functionals on and let {R n },{P n } be corresponding sequences of monic orthogonal polynomials, i.e.,

degR n =degP n =nand1 K n U,R n R m =1 Γ n V,P n P m =δ n,m

where K n 0Γ n and δ n,m is Kronecker delta. The following problem is completely solved: “Describe the pairs of quasi-definite linear functionals U,V for which the differential relation

R n+1 ' (x) n+1+b n R n ' (x) n=P n (x)+a n P n-1 (x)(2)

holds for all positive integer n with b n 0.” The authors prove that at least one of the functionals U,V is semiclassical of class at most 1 if the differential relation (2) holds, and this fact gives a clue to the structure of pairs (U,V).


MSC:
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
33C47Other special orthogonal polynomials and functions