Generalized Hahns theorem

Abstract

Let {P-n(x)}(n=o)(infinity) be an orthogonal polynomial system and [GRAPHICS] a linear differential operator of order k (greater than or equal to 0) with polynomial coefficients. We find necessary and sufficient conditions for a polynomial sequence {Q(n)(x)}(n=o)(infinity) defined by Q(n)(x):=L[P-n+r((r))(x)], n greater than or equal to 0, to be also an orthogonal polynomial system. We also give a few applications of this result together with the complete analysis of the cases: (i) k = 0, 1,2 and r = 0, and (ii) k=r=1. (C) 2000 Elsevier Science B.V. All rights reserved.