On perturbed Szegő recurrences

Abstract

. jm a a . 2 0 1 3 . 1 0 . 0 1 4 © E l s e v i e r 2 0 1 4 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 al L i c e n s e .O n p e r t u r b e d S z e gő r e c u r r e n c e s K e n i e r C a s t i l l o 1 D e p a r t am e n t o d e M a t em á t i c a A p l i c a d a , U n i v e r s i d a d e E s t a d u a l P a u l i s t a -UN E S P , 1 5 0 5 4 -0 0 0 , S ã o J o s é d o R i o P r e t o , S P , B r a z i l abstract K e yw o r d s : T h r e e -t e rm r e c u r r e n c e r e l a t i o n , O r t h o g o n a l p o l y n om i a l s o n t h e r e a l l i n e , S z e g ő r e c u r r e n c e s , O r t h o g o n a l p o l y n om i a l s o n t h e u n i t c i r c l e , P e r t u r b e d V e r b l u n s k y c o effi c i e n t sT h e p u r p o s e o f t h e p r e s e n t c o n t r i b u t i o n i s t o i n v e s t i g a t e t h e e f f e c t s o f fi n i t e m o d i fi c a t i o n s o f V e r b l u n s k y c o effi c i e n t s o n S z e gő r e c u r r e n c e s . M o r e p r e c i s e l y , w e s t u d y t h e s t r u c t u r a l r e l a t i o n s a n d t h e c o r r e s p o n d i n gC -f u n c t i o n s o f t h e o r t h o g o n a l p o l y n om i a l s w i t h r e s p e c t t o t h e s e m o d i fi c a t i o n s f r om t h e i n i t i a l o n e s .1 . I n t r o d u c t i o n . . T h e r e a l l i n e c a s eA s e q u e n c e o f p o l y n om i a l s{ p n ( x ) } n 0 d e fi n e d b y t h e t h r e e -t e rm r e c u r r e n c e r e l a t i o n x p n ( x ) =a n +1 p n +1 ( x ) +b n p n ( x ) +a n p n −1 ( x ) , a n >0 , b n ∈R, ( 1 . 1 ) w i t h i n i t i a l c o n d i t i o n s p −1 ( x ) := 0 a n dp 0 ( x ) := 1 , i s s a i d t o b e a s e q u e n c e o f o r t h o n o rm a l p o l y n om i a l s w i t h r e s p e c t t o a m e a s u r e s u p p o r t e d o n a n i n fi n i t e s u b s e t o f t h e r e a l l i n e I ⊆R. I n d e e d , t h e r e e x i s t s a n o n -t r i s t h e K r o n e c k e r d e l t a . T h i s r e s u l t i s k n ow n i n t h e l i t e r a t u r e o f o r t h o g o n a l p o l y n om i a l s a s F a v a r d ' s t h e o r em [ 3 , 2 6 ] . I n g e n e r a l , t h e o r t h o g o n a l i t y m e a s u r e d μ ( x ) i s n o t u n i q u e , b u t a s uffi c i e n t c o n d i t i o n f o r t h e u n i q u e n e s s i s t h a t t h e r e c u r r e n c ec o effi c i e n t s { a n } n 0 a n d { b n } n 0 a r e b o u n d e d [ 3 ] . T h e a u t h o r t h a n k s t h e r e f e r e e f o r c a r e f u l r e a d i n g a n d v a l u a b l e c omm e n t s . T h e a u t h o r a l s o w i s h e s t o e x p r e s s h i s t h a n k s t o F . M a r c e l l á n f o r s u g g e s t i n g t h e p r o b l em d u r i n g t h e 1 s t J o i n t C o n f e r e n c e o f t h e B e l g i a n , R o y a l S p a n i s h a n d L u x em b o u r g M a t h em a t i c a l S o c i e t i e s i n L i è g e , B e l g i um . T h e r e s e a r c h o f t h e a u t h o r w a s s u p p o r t e d b y CN P q P r o g r...

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

“…The best general reference on this subject is the work of Marcellán et al [33]. More recently, the modification (1.7) has been translated to the theory of OPUC, see [8].…”

Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation

“…Verblunsky theorem states that the OPUC are completely determined by their reflection coefficients. This fact motivated the authors in [6] to study perturbations of Verblunsky coefficients. Again, a transfer matrix approach is used to study the so called co-polynomials on unit circle (COPUC).…”

“…Interlacing properties and some new inequalities involving the zeros of co-modified OPRL, called co-polynomials on real line (COPRL) has been investigated in [8,10]. For details on co-polynomials on unit circle and co-polynomials of R I type, see [6] and [7,28] respectively.…”

“…Remark 2.3. Polynomials obtained after perturbations in (1.1), in (1.2) and in (1.3) have been called co-polynomials on real line (COPRL) [8], co-polynomials on unit circle (COPUC) [6] and co-polynomials of R I type respectively [28]. Following an analogous nomenclature, we may call perturbed polynomials introduced Section 2 as co-polynomials of R II type.…”

On co-polynomials on the real line

“…Moreover, we will consider the finite composition of the above perturbations. In Section 2, we study some new inequalities for the zeros of COPRL by following the approach presented in [5] for the study of the monotonicity of zeros of a class of para-orthogonal polynomials on the unit circle including the Askey hypergeometric polynomials 2 F 1 (−n, a + bi; 2a; 1 − z), a, b ∈ R. In Section 3, we obtain a new structural relation based on a transfer matrix approach proposed recently in [4] for similar perturbations in the theory of orthogonal polynomials on the unit circle (OPUC, in short). Finally, in Section 3 we point out the connection with the OPUC.…”