Sieved orthogonal polynomials. VII. Generalized polynomial mappings

Charris, Jairo A.; Ismail, Mourad E. H.

doi:10.1090/S0002-9947-1993-1038014-4

Sieved orthogonal polynomials. VII. Generalized polynomial mappings
HTML articles powered by AMS MathViewer

by Jairo A. Charris and Mourad E. H. Ismail PDF

Trans. Amer. Math. Soc. 340 (1993), 71-93 Request permission

Abstract:

Systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second-order difference equations are studied in detail. Applications are given to the theory of the recently discovered sieved orthogonal polynomials. The connection with polynomial mappings is examined. An example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included.

References

Waleed Al-Salam, W. R. Allaway, and Richard Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 284 (1984), no. 1, 39–55. MR 742411, DOI 10.1090/S0002-9947-1984-0742411-6

The Rogers

ultraspherical polynomials

R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
Richard Askey, Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum, SIAM J. Math. Anal. 5 (1974), 119–124. MR 385197, DOI 10.1137/0505013
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
Joaquin Bustoz and Mourad E. H. Ismail, The associated ultraspherical polynomials and their $q$-analogues, Canadian J. Math. 34 (1982), no. 3, 718–736. MR 663314, DOI 10.4153/CJM-1982-049-6
Jairo Charris and Mourad E. H. Ismail, On sieved orthogonal polynomials. II. Random walk polynomials, Canad. J. Math. 38 (1986), no. 2, 397–415. MR 833576, DOI 10.4153/CJM-1986-020-x
Jairo A. Charris and Mourad E. H. Ismail, On sieved orthogonal polynomials. V. Sieved Pollaczek polynomials, SIAM J. Math. Anal. 18 (1987), no. 4, 1177–1218. MR 892496, DOI 10.1137/0518086
Jairo A. Charris and Guillermo Rodríguez-Blanco, On systems of orthogonal polynomials with inner and end point masses, Rev. Colombiana Mat. 24 (1990), no. 3-4, 153–177 (Spanish). MR 1106603
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
J. S. Geronimo and W. Van Assche, Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986), no. 3, 251–283. MR 840395, DOI 10.1016/0021-9045(86)90065-1
J. S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), no. 2, 559–581. MR 951620, DOI 10.1090/S0002-9947-1988-0951620-6
Mourad E. H. Ismail, On sieved orthogonal polynomials. I. Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), no. 5, 1093–1113. MR 800799, DOI 10.1137/0516081
Jairo Charris and Mourad E. H. Ismail, On sieved orthogonal polynomials. II. Random walk polynomials, Canad. J. Math. 38 (1986), no. 2, 397–415. MR 833576, DOI 10.4153/CJM-1986-020-x
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697

On the expansion of some infinite products

24

Second memoir on the expansion of some infinite products

25

Third memoir on the expansion of some infinite products

26

H. A. Slim, On co-recursive orthogonal polynomials and their application to potential scattering, J. Math. Anal. Appl. 136 (1988), no. 1, 1–19. MR 972579, DOI 10.1016/0022-247X(88)90111-4

Modified moments and continued fraction coefficients for the diatomic linear chain

80