Hahn%20polynomials

(0.001 seconds)

11—20 of 316 matching pages

11: 18.20 Hahn Class: Explicit Representations

§18.20 Hahn Class: Explicit Representations

►

§18.20(i) Rodrigues Formulas

… ►For the Hahn polynomials

p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right)

and … ►

Continuous Hahn

… ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

…

12: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

… ►

Hahn

►

18.23.1

{{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

►

18.23.2

{{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

… ►

Continuous Hahn

…

13: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

Hahn

… ►

Continuous Hahn

… ►

Hahn

… ►

§18.22(iii) $x$ -Differences

►

Hahn

…

15: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

►Ismail (1986) gives asymptotic expansions as

n\to\infty

, with

x

and other parameters fixed, for continuous

q

-ultraspherical, big and little

q

-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson

p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)

the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the

q

-Laguerre and continuous

q^{-1}

-Hermite polynomials see Chen and Ismail (1998).

16: 18.38 Mathematical Applications

… ►

Approximation Theory

… ►

Integrable Systems

… ►The

3j

symbol (34.2.6), with an alternative expression as a terminating

{{}_{3}F_{2}}

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. The orthogonality relations in §34.3(iv) for the

3j

symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. … …

17: Bibliography K

… ►

S. Karlin and J. L. McGregor (1961) The Hahn polynomials, formulas and an application. Scripta Math. 26, pp. 33–46.

ⓘ

External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §18.20(i), §18.21(ii), §18.26(ii).
Encodings:: BibTeX

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1981) Clebsch-Gordan coefficients for

{\rm SU}(2)

and Hahn polynomials. Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.

ⓘ

External Links:: ISSN 0028-9825, MathReview (Jiří Patera), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1984b) Orthogonal polynomials with weight function

(1-x)^{\alpha}(1+x)^{\beta}+M\delta(x+1)+N\delta(x-1)

. Canad. Math. Bull. 27 (2), pp. 205–214.

ⓘ

External Links:: ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt
Referenced by:: §18.36(i), §18.36(i).
Encodings:: BibTeX

…

18: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

R. Askey (1985) Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

ⓘ

External Links:: ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.19, §18.20(ii), §18.21(ii).
Encodings:: BibTeX

…

19: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

20: Bibliography L

… ►

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.

ⓘ

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

… ►

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

ⓘ

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

… ►

Y. Lin and R. Wong (2013) Global asymptotics of the Hahn polynomials. Anal. Appl. (Singap.) 11 (3), pp. 1350018, 47.

ⓘ

External Links:: ISSN 0219-5305, Document, FullText, MathReview (Yamilet Quintana), ZentralBlatt
Referenced by:: §18.24, §18.24.
Encodings:: BibTeX

… ►

J. L. López and N. M. Temme (1999a) Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview, ZentralBlatt
Referenced by:: §18.15(vi), §18.16(vi).
Encodings:: BibTeX

►

J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt
Referenced by:: §18.34(iii), §24.11, §24.16(i), §24.16(i).
Encodings:: BibTeX

…

Hahn%20polynomials

11—20 of 316 matching pages

11: 18.20 Hahn Class: Explicit Representations

§18.20 Hahn Class: Explicit Representations

§18.20(i) Rodrigues Formulas

Continuous Hahn

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

12: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

Hahn

Continuous Hahn

13: 18.22 Hahn Class: Recurrence Relations and Differences

Hahn

Continuous Hahn

Hahn

§18.22(iii) $x$ -Differences

Hahn

14: 18.27 $q$ -Hahn Class

§18.27 $q$ -Hahn Class

§18.27(ii) $q$ -Hahn Polynomials

From Big $q$ -Jacobi to Jacobi

From Big $q$ -Jacobi to Little $q$ -Jacobi

Limit Relations

15: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

16: 18.38 Mathematical Applications

Approximation Theory

Integrable Systems

17: Bibliography K

18: Bibliography

19: Wolter Groenevelt

20: Bibliography L

Hahn%20polynomials

11—20 of 316 matching pages

11: 18.20 Hahn Class: Explicit Representations

§18.20 Hahn Class: Explicit Representations

§18.20(i) Rodrigues Formulas

Continuous Hahn

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

12: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

Hahn

Continuous Hahn

13: 18.22 Hahn Class: Recurrence Relations and Differences

Hahn

Continuous Hahn

Hahn

§18.22(iii) x -Differences

Hahn

14: 18.27 q -Hahn Class

§18.27 q -Hahn Class

§18.27(ii) q -Hahn Polynomials

From Big q -Jacobi to Jacobi

From Big q -Jacobi to Little q -Jacobi

Limit Relations

15: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes

16: 18.38 Mathematical Applications

Approximation Theory

Integrable Systems

17: Bibliography K

18: Bibliography

19: Wolter Groenevelt

20: Bibliography L

§18.22(iii) $x$ -Differences

14: 18.27 $q$ -Hahn Class

§18.27 $q$ -Hahn Class

§18.27(ii) $q$ -Hahn Polynomials

From Big $q$ -Jacobi to Jacobi

From Big $q$ -Jacobi to Little $q$ -Jacobi

15: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes