Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

Abstract

We consider the Sobolev inner product 〈 f , g 〉 = ∫ − 1 1 f ( x ) g ( x ) ( 1 − x 2 ) α − 1 2 d x + ∫ f ′ ( x ) g ′ ( x ) d ψ ( x ) , α > − 1 2 , where d ψ is a measure involving a Gegenbauer weight and with mass points outside the interval ( − 1 , 1 ) . We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler–Heine type formula. These results are illustrated with some numerical experiments.