ABSTRACT
In this work, we will study the family of polynomials which are orthogonal with respect to the Sobolev inner product: ( f , g ) s ( N ) = ∑ m , k = 0 N 〈 λ m , k u , f ( m ) g ( k ) 〉 N ≥ 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3438.tif"/> where u is a semiclassical positive definite linear functional and λ m, k = λ m, k (x) are polynomials for m, k = 0,…, N, such that Λ N = ( λ m , k ( x ) ) m , k = 0 N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3439.tif"/> symmetric and positive definite matrix. For this inner product, we deduce the expression of a symmetric linear operator which plays an essential role in the relation between the orthogonal polynomials associated to the linear functional u and the orthogonal polynomials associated to ( . , . ) S ( N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3440.tif"/> . Also, for these non-standard orthogonal polynomials, a difference-differential relation is deduced.
AMS Subject Classification (1991): 33 C 45
