## Sieved orthogonal polynomials. VII. Generalized polynomial mappings

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- 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.Â 1, 39â€“55. MR**742411**, DOI 10.1090/S0002-9947-1984-0742411-6
R. Aksey and M. E. H. Ismail, - R. Askey and Mourad E. H. Ismail,
*A generalization of ultraspherical polynomials*, Studies in pure mathematics, BirkhĂ¤user, Basel, 1983, pp.Â 55â€“78. MR**820210** - Richard Askey and Mourad Ismail,
*Recurrence relations, continued fractions, and orthogonal polynomials*, Mem. Amer. Math. Soc.**49**(1984), no.Â 300, iv+108. MR**743545**, DOI 10.1090/memo/0300 - Richard Askey,
*Jacobi polynomials. I. New proofs of Koornwinderâ€™s Laplace type integral representation and Batemanâ€™s bilinear sum*, SIAM J. Math. Anal.**5**(1974), 119â€“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.Â 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.Â 3, 718â€“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.Â 2, 397â€“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.Â 4, 1177â€“1218. MR**892496**, DOI 10.1137/0518086 - Jairo A. Charris and Guillermo RodrĂguez-Blanco,
*On systems of orthogonal polynomials with inner and end point masses*, Rev. Colombiana Mat.**24**(1990), no.Â 3-4, 153â€“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.Â 3, 251â€“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.Â 2, 559â€“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.Â 5, 1093â€“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.Â 2, 397â€“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**
L. J. Rogers, - H. A. Slim,
*On co-recursive orthogonal polynomials and their application to potential scattering*, J. Math. Anal. Appl.**136**(1988), no.Â 1, 1â€“19. MR**972579**, DOI 10.1016/0022-247X(88)90111-4
J. C. Wheeler,

*The Rogers*$q$-

*ultraspherical polynomials*, Approximation Theory III, Academic Press, 1980, pp. 223-278.

*On the expansion of some infinite products*, Proc. London Math. Soc. (3)

**24**(1892), 337-352. â€”,

*Second memoir on the expansion of some infinite products*, Proc. London Math. Soc. (3)

**25**(1894), 318-342. â€”,

*Third memoir on the expansion of some infinite products*, Proc. London Math. Soc. (3)

**26**(1895), 15-32.

*Modified moments and continued fraction coefficients for the diatomic linear chain*, J. Chem. Phys.

**80**(1984), 472-476.

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**340**(1993), 71-93 - MSC: Primary 33C45
- DOI: https://doi.org/10.1090/S0002-9947-1993-1038014-4
- MathSciNet review: 1038014