Two systems of sieved Jacobi polynomials introduced by R. Askey are considered. Their orthogonality measures are determined via the theory of blocks of recurrence relations, circumventing any resort to properties of the Askey-Wilson polynomials. The connection with polynomial mappings is examined. Some naturally related systems are also dealt with and a simple procedure to compute their orthogonality measures is devised which seems to be applicable in many other instances.
@article{bwmeta1.element.bwnjournal-article-cmv78z1p57bwm,
author = {Bernarda Aldana and Jairo Charris and Oriol Mora-Valbuena},
title = {On block recursions, Askey's sieved Jacobi polynomials and two related systems},
journal = {Colloquium Mathematicae},
volume = {78},
year = {1998},
pages = {57-91},
zbl = {0961.33007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv78z1p57bwm}
}
Aldana, Bernarda; Charris, Jairo; Mora-Valbuena, Oriol. On block recursions, Askey's sieved Jacobi polynomials and two related systems. Colloquium Mathematicae, Tome 78 (1998) pp. 57-91. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv78z1p57bwm/
[000] [1] N. Al-Salam and M. E. H. Ismail, On sieved orthogonal polynomials VIII: Sieved associated Pollaczek polynomials, J. Approx. Theory 68 (1992), 306-321. | Zbl 0746.33001
[001] [2] W. Al-Salam, W. Allaway and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 234 (1984), 39-55.
[002] [3] G. Andrews, q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conf. Ser. in Math. 66, Amer. Math. Soc., Providence, R.I., 1986.
[003] [4] R. Askey, Orthogonal polynomials old and new, and some combinatorial connections, in: Enumeration and Design, D. M. Jackson and S. A. Vanstone (eds.), Academic Press, Toronto, Ont., 1984, 67-84.
[004] [5] R. Askey and M. E. H. Ismail, The Rogers q-ultraspherical polynomials, in: Approximation Theory III, E. W. Cheney (ed.), Academic Press, New York, N.Y., 1980, 175-182. | Zbl 0479.33013
[005] [6] R. Askey and M. E. H. Ismail, A generalization of the ultraspherical polynomials, in: Studies in Pure Mathematics, P. Erdős (ed.), Birkhäuser, Basel, 1983, 55-78. | Zbl 0532.33006
[006] [7] R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300 (1984). | Zbl 0548.33001
[007] [8] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319 (1985). | Zbl 0572.33012
[008] [9] S. Belmehdi, On the left multiplication of a regular linear functional by a polynomial, in: Orthogonal Polynomials and Their Applications, IMACS Ann. Comput. Appl. Math. 9, Baltzer, Basel, 1991, 169-175. | Zbl 0837.42013
[009] [10] J. A. Charris and L. A. Gómez, On the orthogonality measure of the q-Pollaczek polynomials, Rev. Colombiana Mat. 21 (1987), 301-316. | Zbl 0665.33010
[010] [11] J. A. Charris and M. E. H. Ismail, On sieved orthogonal polynomials II: Random walk polynomials, Canad. J. Math. 38 (1986), 397-415. | Zbl 0576.33006
[011] [12] J. A. Charris and M. E. H. Ismail, On sieved orthogonal polynomials V: Sieved Pollaczek polynomials, SIAM J. Math. Anal. 18 (1987), 1177-1218. | Zbl 0645.33018
[012] [13] J. A. Charris and M. E. H. Ismail, Sieved orthogonal polynomials VII: Generalized polynomial mappings, Trans. Amer. Math. Soc. 340 (1993), 71-93. | Zbl 0790.33007
[013] [14] J. A. Charris, M. E. H. Ismail and S. Monsalve, Sieved orthogonal polynomials X: General blocks of recurrence relations, Pacific J. Math. 163 (1994), 1294-1308. | Zbl 0805.33006
[014] [15] J. A. Charris and Y. L. Prieto, On distributional representation of moment functionals: Sieved Pollaczek polynomials, Rev. Acad. Colombiana Cienc. 73 (1994), 305-315.
[015] [16] J. A. Charris and F. Soriano, Complex and distributional weights for sieved ultraspherical polynomials, Internat. J. Math. Math. Sci. 19 (1996), 229-242. | Zbl 0905.33004
[016] [17] J. A. Charris and F. Soriano, On the distributional orthogonality of the general Pollaczek polynomials, ibid., 417-426. | Zbl 0908.33007
[017] [18] T. S. Chihara, On co-recursive orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957), 899-905. | Zbl 0080.27305
[018] [19] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, N.Y., 1978. | Zbl 0389.33008
[019] [20] T. S. Chihara, On kernel polynomials and related systems, Boll. Un. Mat. Ital. (3) 19 (1964), 451-459 | Zbl 0125.31406
[020] [21] J. Fields, A unified treatment of Darboux's method, Arch. Rational. Mech. Anal. 27 (1967/68), 289-305. | Zbl 0183.32201
[021] [22] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. | Zbl 0695.33001
[022] [23] J. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), 559-581. | Zbl 0652.42009
[023] [24] M. E. H. Ismail, Sieved orthogonal polynomials I: Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 89-111. | Zbl 0576.33005
[024] [25] M. E. H. Ismail, Sieved orthogonal polynomials III: Orthogonality on several intervals, Trans. Amer. Math. Soc. 294 (1986), 89-111. | Zbl 0612.33009
[025] [26] M. E. H. Ismail and D. R. Masson, Two families of orthogonal polynomials related to the Jacobi polynomials, Rocky Mountain J. Math. 21 (1991), 359-375. | Zbl 0744.33004
[026] [27] M. E. H. Ismail, D. Masson and M. Rahman, Complex weight functions for classical orthogonal polynomials, Canad. J. Math. 43 (1991), 1294-1308. | Zbl 0752.33003
[027] [28] S. Lang, Real and Functional Analysis, 3rd ed., Springer, New York, N.Y., 1993. | Zbl 0831.46001
[028] [29] F. Marcellan, J. S. Dehesa and A. Ronveaux, On orthogonal polynomials with perturbed recurrence relations, J. Comput. Appl. Math. 30 (1990), 203-212. | Zbl 0713.42021
[029] [30] F. W. Olver, Asymptotics and Special Functions, Academic Press, New York, N.Y., 1974. | Zbl 0303.41035
[030] [31] E. D. Rainville, Special Functions, Macmillan, New York, N.Y., 1960.
[031] [32] L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337-352. | Zbl 25.0432.01
[032] [33] L. J. Rogers, Second memoir on the expansion of some infinite products, ibid. 25 (1894), 318-342.
[033] [34] L. J. Rogers, Third memoir on the expansion of some infinite products, ibid. 26 (1895), 15-32. | Zbl 26.0289.01
[034] [35] L. J. Rogers, On two theorems of combinatory analysis and some allied identities, ibid. 16 (1917), 315-336. | Zbl 46.0109.01
[035] [36] L. J. Rogers and S. Ramanujan, Proof of certain identities in combinatory analysis, Proc. Cambridge Philos. Soc. 19 (1919), 211-216. | Zbl 47.0903.01
[036] [37] H. A. Slim, On co-recursive orthogonal polynomials and their application to potential scattering, J. Math. Anal. Appl. 136 (1988), 1-19. | Zbl 0672.42017
[037] [38] H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice Hall, Englewood Cliffs, N.J., 1966.
[038] [39] G. Szegő, Ein Beitrag zur Theorie der Polynome von Laguerre und Jacobi, Math. Z. 1 (1918), 341-456. | Zbl 46.0583.04
[039] [40] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, R.I., 1975.
[040] [41] H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, N.Y., 1948. | Zbl 0035.03601