We consider the following discrete Sobolev inner product involving the Gegenbauer weight (f, g) S :=

In this paper, we introduce varying generalized Freud-type polynomials which are orthogonal with respect to a varying discrete Freud-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as find a second-order differential–difference equation that these polynomials satisfy. To reach this objective, it is necessary to consider the standard Freud orthogonal polynomials and, in the meanwhile, we find new difference relations for the coefficients in the first-order differential equations that this standard family satisfies.

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently.

M a ñ a s , J . F . , M a r c e l l á n , F . , & M o r e n o , J . J . ( 2 0 1 3 ) . V a r y i n g d i s c r e t e L a g u e r r e S o b o l e v o r t h o g o n a l p o l y n om i a l s : a s ym p t o t i c b e h a v i o r a n d z e r o s . A p p l i e d M a t h em a t i c s a n d C om p u t a t i o n, 2 2 2 , p p . 6 1 2 -6 1 8 . DO I : 1 0 . 1 0 1 6 / j . am c . 2 0 1 3 . 0 7 . 0 7 4 P r o y e c t o s : M TM 2 0 1 1 2 8 9 5 2 C 0 2 0 1 M TM 2 0 1 2 3 6 7 3 2 C 0 3 0 1 © E l s e v i e r 2 0 1 3 T h i s w o r k i s l i c e n s e d u n d e r a C r e a t i v e C omm o n s A t t r i b u t i o n -N o n C omm e r c i a l -N oD e r i v a t i v e s 4 . 0 I n t e r n a t i o n a l L i c e n s e .

No abstract