zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Structure relations for orthogonal polynomials on the unit circle. (English) Zbl 1242.33011
Summary: Structure relations for orthogonal polynomials with respect to Hermitian linear functionals are studied. Firstly, we prove that semi-classical orthogonal polynomials satisfy structure relations of the following type: $$\sum^{s_1}_{k=0}\beta_{n,k}P_{n+s_1-k}+\sum^{s_2}_{k=0}\gamma_{n,k}z^kP^*_{n-1-k}=\sum^{r_1}_{k=0}\alpha_{n,k}P^{[1]}_{n+s_1-k}+\sum^{r_2}_{k=0}\eta_{n,k}(P^*_{n+r_2-k})',$$ where $s_1,s_2,r_1,r_2$ are integers (specified in the text), $P^*_n$ is the reversed polynomial of $P_n,P^{[1]}_n=P_{n+1}'/(n+1)$, and $\beta_{n,k},\gamma_{n,k},\alpha_{n,k},\eta_{n,k}$ are complex numbers. Then, we study the semi-classical character of sequences of orthogonal polynomials $\{R_n\},\{P_n\}$, connected through a structure relation of the following type: $$\sum^{s_1}_{k=0}\beta_{n,k}R_{n+s_1-k}+\sum^{s_2}_{k=0}\gamma_{n,k}R^*_{n +s_2-k}=\sum^{r_1}_{k=0}\alpha_{n,k}P^{[1]}_{n+r_1-k}+\sum^{r_2}_{k=0}\eta_{n,k}(P^*_{n+r_2-k})',$$ where the integers $s_1,s_2,r_1,r_2$ satisfy some natural conditions specified in the text.

33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
[1] Alfaro, M.; Moral, L.: Quasi-orthogonality on the unit circle and semi-classical forms, Portugal math. 51, No. 1, 47-62 (1994) · Zbl 0812.42013
[2] Bonan, S.; Lubinsky, D.; Nevai, P.: Orthogonal polynomials and their derivatives II, SIAM J. Math. anal. 18, 1163-1176 (1987) · Zbl 0638.42023 · doi:10.1137/0518085
[3] Branquinho, A.: A note on semi-classical orthogonal polynomials, Bull. belg. Math. soc. 3, 1-12 (1996) · Zbl 0862.42018
[4] Branquinho, A.; Moreno, A. Foulquié; Marcellán, F.; Rebocho, M. N.: Coherent pairs of linear functionals on the unit circle, J. approx. Theory 153, No. 1, 122-137 (2008) · Zbl 1149.42013 · doi:10.1016/j.jat.2008.03.003
[5] Branquinho, A.; Golinskii, L.; Marcellán, F.: Orthogonal polynomials and rational modifications of Lebesgue measures on the unit circle. An inverse problem, Complex var. Theory appl. 38, No. 2, 137-154 (1999) · Zbl 1022.42015
[6] Branquinho, A.; Rebocho, M. N.: On the semiclassical character of orthogonal polynomials satisfying structure relations, J. difference equ. Appl. 18, 111-138 (2012) · Zbl 1241.33012
[7] Cantero, M. J.; Moral, L.; Velázquez, L.: Direct and inverse polynomial perturbations of Hermitian linear functionals, J. approx. Theory 163, No. 8, 988-1028 (2011) · Zbl 1221.42045 · doi:10.1016/j.jat.2011.02.014
[8] Castillo, K.; Garza, L.; Marcellán, F.: Perturbations on the subdiagonals of Toeplitz matrices, Linear algebra appl. 434, 1563-1579 (2011) · Zbl 1208.42010 · doi:10.1016/j.laa.2010.11.037
[9] Ya L. Geronimus, Polynomials Orthogonal on a Circle and their Applications, Amer. Math. Soc., Providence, RI, 1962.
[10] Godoy, E.; Marcellán, F.: Tridiagonal Toeplitz matrices and orthogonal polynomials on the unit circle, Orthogonal polynomials and their applications (Laredo, 1987), lecture notes in pure and appl. Math. 117, 139-146 (1989) · Zbl 0673.42013
[11] Golinskii, L.; Nevai, P.: Szegő difference equations transfer matrices and orthogonal polynomials on the unit circle, Commun. math. Phys. 223, 223-259 (2001) · Zbl 0998.42015 · doi:10.1007/s002200100525
[12] De Jesus, M. N.; Petronilho, J.: On linearly related sequences of derivatives of orthogonal polynomials, J. math. Anal. appl. 347, 482-492 (2008) · Zbl 1160.42011 · doi:10.1016/j.jmaa.2008.06.017
[13] Kwon, K. H.; Lee, J. H.; Marcellán, F.: Generalized coherent pairs, J. math. Anal. appl. 253, No. 2, 482-514 (2001) · Zbl 0967.33005 · doi:10.1006/jmaa.2000.7157
[14] A. Magnus, Semi-classical orthogonal polynomials on the unit circle, 2000. Available from: <http://perso.uclouvain.be/alphonse.magnus/MAPA3072A>.
[15] Marcellán, F.; Maroni, P.: Orthogonal polynomials on the unit circle and their derivatives, Constr. approx. 7, 341-348 (1991) · Zbl 0734.42015 · doi:10.1007/BF01888162
[16] Marcellán, F.; Peherstorfer, F.; Steinbauer, R.: Orthogonality properties of linear combinations of orthogonal polynomials II, Adv. comput. Math. 7, 401-428 (1997) · Zbl 0933.42012 · doi:10.1023/A:1018963323132
[17] Marcellán, F.; Sfaxi, R.: Second structure relation for semiclassical orthogonal polynomials, J. comput. Appl. math. 200, 537-554 (2007) · Zbl 1125.33008 · doi:10.1016/j.cam.2006.01.007
[18] P. Maroni, Une théorie algébrique des polynômes orthogonaux, Application aux polynômes orthogonaux semi-classiques, in: C. Brezinski et al. (Eds.), Orthogonal Polynomials and their Applications, Annals Comput. Appl. Math. 9, Baltzer, Basel, 1991, pp. 95 -- 130. · Zbl 0944.33500
[19] Finkelshtein, A. Martínez: Analytic aspects of Sobolev orthogonal polynomials, J. comput. Appl. math. 127, 255-266 (2001) · Zbl 0971.33004 · doi:10.1016/S0377-0427(00)00499-4
[20] Meijer, H. G.: Determination of all coherent pairs, J. approx. Theory 89, 321-343 (1997) · Zbl 0880.42012 · doi:10.1006/jath.1996.3062
[21] Shohat, J. A.: On mechanical quadratures in particular with positive coefficients, Trans. amer. Math. soc. 42, 461-496 (1937) · Zbl 0018.11902 · doi:10.2307/1989740
[22] G. Szegő, Orthogonal polynomials, in: AMS Colloq. Publ., fourth ed., vol. 23, AMS, Providence, RI, 1975.
[23] C. Tasis, Propiedades diferenciales de los polinomios ortogonales relativos a la circunferencia unidad, Doctoral Dissertation, Universidad de Cantabria, 1989.