Characterizations of generalized Hermite and sieved ultraspherical polynomials

Dette, Holger

doi:10.1090/S0002-9947-96-01438-9

Characterizations of generalized Hermite and sieved ultraspherical polynomials
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Trans. Amer. Math. Soc. 348 (1996), 691-711 Request permission

Abstract:

A new characterization of the generalized Hermite polyno- mials and of the orthogonal polynomials with respect to the measure $|x|^\gamma (1-x^2)^{1/2}dx$ is derived which is based on a “reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.

References

M. Abramowitz and I. Stegun. Handbook of mathematical functions, Dover, New York, 1964.
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
W. A. Al-Salam, Characterization theorems for orthogonal polynomials, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 1–24. MR 1100286, DOI 10.1007/978-94-009-0501-6_{1}
Richard Askey, Orthogonal polynomials old and new, and some combinatorial connections, Enumeration and design (Waterloo, Ont., 1982) Academic Press, Toronto, ON, 1984, pp. 67–84. MR 782309
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, Sieved orthogonal polynomials. VII. Generalized polynomial mappings, Trans. Amer. Math. Soc. 340 (1993), no. 1, 71–93. MR 1038014, DOI 10.1090/S0002-9947-1993-1038014-4
Jairo A. Charris, Mourad E. H. Ismail, and Sergio Monsalve, On sieved orthogonal polynomials. X. General blocks of recurrence relations, Pacific J. Math. 163 (1994), no. 2, 237–267. MR 1262296, DOI 10.2140/pjm.1994.163.237
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
H. Dette and W. J. Studden, On a new characterization of the classical orthogonal polynomials, J. Approx. Theory 71 (1992), no. 1, 3–17. MR 1180871, DOI 10.1016/0021-9045(92)90128-B
Wolfgang Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials $L_n^{(an+\alpha )}$ and Hermite polynomials $H_n^{(an+\alpha )}$, Analysis 13 (1993), no. 1-2, 29–67. MR 1245742, DOI 10.1524/anly.1993.13.12.29
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
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
Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
Tai-Shing Lau and W. J. Studden, On an extremal problem of Fejér, J. Approx. Theory 53 (1988), no. 2, 184–194. MR 945871, DOI 10.1016/0021-9045(88)90065-2
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15–32.
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Walter Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987. MR 903848, DOI 10.1007/BFb0081880
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336