DLMF: Untitled Document

§18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Jacobi

… ►A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1. …

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A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:: ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

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M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

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External Links:: ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.26(iv).
Encodings:: BibTeX

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M. E. H. Ismail and D. R. Masson (1994)

q

-Hermite polynomials, biorthogonal rational functions, and

q

-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:: ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.28(vii).
Encodings:: BibTeX

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M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and

q

-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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External Links:: ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §18.29.
Encodings:: BibTeX

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M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

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L. Habsieger (1988) Une

q

-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:: ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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E. Hahn (1980) Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen. Math. Z. 171 (3), pp. 201–226 (German).

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External Links:: ISSN 0025-5874, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.15(i).
Encodings:: BibTeX

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F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

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CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:: Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).
External Links:: CAOP
Referenced by:: 1st item.
Encodings:: BibTeX

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B. C. Carlson (1985) The hypergeometric function and the

R

-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:: International conference on special functions: theory and computation (Turin, 1984)
External Links:: ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.27(vi).
Encodings:: BibTeX

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L. Chihara (1987) On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.

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External Links:: ISSN 0036-1410, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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D. V. Chudnovsky and G. V. Chudnovsky (1988) Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.

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External Links:: MathReview (Maurice Mignotte), ZentralBlatt
Referenced by:: §20.11(iii).
Encodings:: BibTeX

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D. R. Masson (1991) Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.

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External Links:: ISSN 0176-4276, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

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J. Meixner, F. W. Schäfke, and G. Wolf (1980) Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies. Lecture Notes in Mathematics, Vol. 837, Springer-Verlag, Berlin-New York.

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External Links:: ISBN 3-540-10282-5, 0-387-10282-5, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §28.10(ii), §28.11, §28.2(v), §28.27, §28.28(i), §28.28(v), §28.30(ii), §28.6(i), §28.6(i), §28.6(i), §28.7, §28.7, Ch.28, §30.11(vi), §30.12, §30.13(v), §30.14(v), §30.15(i), §30.3(iv), Ch.30, Profile Gerhard Wolf.
Encodings:: BibTeX

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D. S. Mitrinović (1970) Analytic Inequalities. Springer-Verlag, New York.

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Notes:: Addenda: Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., no. 634–677 (1979), pp. 3–24 and no. 577-598 (1977), pp. 3–10.
External Links:: MathReview (R. A. Askey and R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §4.18, §4.32, §4.5(i), §4.5(ii), §7.8.
Encodings:: BibTeX

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D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

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J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

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N. M. Temme and J. L. López (2001) The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.

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External Links:: ISSN 0377-0427, Document, MathReview (P. N. Rathie), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:: ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt
Referenced by:: §13.8(iii), §13.8(iii).
Encodings:: BibTeX

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P. Terwilliger (2011) The universal Askey-Wilson algebra. SIGMA 7, pp. Paper 069, 24 pp..

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External Links:: ISSN 1815-0659, Link, MathReview Entry, ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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P. Terwilliger (2013) The universal Askey-Wilson algebra and DAHA of type

(C^{\vee}_{1},C_{1})

. SIGMA 9, pp. Paper 047, 40 pp..

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External Links:: ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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D. F. Lawden (1989) Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:: ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.
Encodings:: BibTeX

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P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

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J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:: Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II
External Links:: ISSN 1073-2772, Pdf, MathReview, ZentralBlatt
Referenced by:: §13.21(i), §13.24(ii).
Encodings:: BibTeX

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D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:: MathReview (J. Browkin), ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

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Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:: Includes hypergeometric and $q$ -hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).
Encodings:: BibTeX

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J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:: ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

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A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

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External Links:: MathReview (Vadim B. Kuznetsov), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

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External Links:: ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.33(iii).
Encodings:: BibTeX

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Askey%E2%80%93Wilson%20class%20orthogonal%20polynomials

11—20 of 377 matching pages

11: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

§18.21(i) Dualities

§18.21(ii) Limit Relations and Special Cases

Hahn $\to$ Jacobi

12: Bibliography I

13: Bibliography H

14: 5 Gamma Function

15: 16 Generalized Hypergeometric Functions & Meijer G-Function

16: Bibliography C

17: Bibliography M

18: Bibliography T

19: Bibliography L

20: Bibliography Z

Askey%E2%80%93Wilson%20class%20orthogonal%20polynomials

11—20 of 377 matching pages

11: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

§18.21(i) Dualities

§18.21(ii) Limit Relations and Special Cases

Hahn → Jacobi

12: Bibliography I

13: Bibliography H

14: 5 Gamma Function

15: 16 Generalized Hypergeometric Functions & Meijer G-Function

16: Bibliography C

17: Bibliography M

18: Bibliography T

19: Bibliography L

20: Bibliography Z

Hahn $\to$ Jacobi