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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:

k=0 s 1 β n,k P n+s 1 -k + k=0 s 2 γ n,k z k P n-1-k * = k=0 r 1 α n,k P n+s 1 -k [1] + k=0 r 2 η 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 n [1] =P n+1 ' /(n+1), and β n,k ,γ n,k ,α n,k ,η 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:

k=0 s 1 β n,k R n+s 1 -k + k=0 s 2 γ n,k R n+s 2 -k * = k=0 r 1 α n,k P n+r 1 -k [1] + k=0 r 2 η 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.

MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
References:
[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)
[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: lt;http://perso.uclouvain.be/alphonse.magnus/MAPA3072Agt;.
[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.