Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics

Abstract

We consider the following discrete Sobolev inner product involving the Gegenbauer weight (f,g)S≔∫−11f(x)g(x)(1−x2)αdx+M[f(j)(−1)g(j)(−1)+f(j)(1)g(j)(1)],where α>−1,j∈N∪{0}, and M>0. Our main objective is to calculate the exact value r0=limn→+∞logmaxx∈[−1,1]|Q˜n(α,M,j)(x)|logλ˜n,α≥−1∕2,where {Q˜n(α,M,j)}n≥0 is the sequence of orthonormal polynomials with respect to this Sobolev inner product. These polynomials are eigenfunctions of a differential operator and the obtaining of the asymptotic behavior of the corresponding eigenvalues, λ˜n, is the principal key to get the result. This value r0 is related to the convergence of a series in a left-definite space. In addition, to complete the asymptotic study of this family of nonstandard polynomials we give the Mehler–Heine formulae for the corresponding orthogonal polynomials.