Krawtchouk%20polynomials

SearchAdvancedHelp

(0.002 seconds)

11—20 of 316 matching pages

11: 18.1 Notation

… ►

Classical OP’s

… ►

Hahn Class OP’s

… ►

Krawtchouk: $K_n(x;p,N)$.

… ►

Wilson Class OP’s

… ►Nor do we consider the shifted Jacobi polynomials: …

12: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

… ►

Hahn

… ►

Krawtchouk

►

18.23.3 $${\left(1-\frac{1-p}{p}z\right)}^x{(1+z)}^{N-x}=\sum _{n=0}^N\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)K_n(x;p,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.7 $${(1-{\mathrm{e}}^{\mathrm{i}\varphi }z)}^{-\lambda +\mathrm{i}x}{(1-{\mathrm{e}}^{-\mathrm{i}\varphi }z)}^{-\lambda -\mathrm{i}x}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(x;\varphi )z^n,$$ .

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

13: 18.38 Mathematical Applications

… ►

Approximation Theory

… ►

Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. … ►

Coding Theory

►For applications of Krawtchouk polynomials $K_n(x;p,N)$ and $q$-Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). …

14: 18.26 Wilson Class: Continued

… ►

§18.26(ii) Limit Relations

… ►

Dual Hahn $\to$ Krawtchouk

… ►See also Figure 18.21.1. ►

§18.26(iii) Difference Relations

… ►

§18.26(v) Asymptotic Approximations

…

15: Bibliography L

… ►

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

… ►

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

… ►

A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory 106 (1), pp. 155–184.

ⓘ

External Links:

ISSN 0021-9045, Document, MathReview (Ahmed I. Zayed), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

… ►

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

►

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

…

16: 15.9 Relations to Other Functions

… ►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

Legendre

… ►

Krawtchouk

… ►

Meixner

…

17: 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. …

18: William P. Reinhardt

… ►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. …Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

19: 20 Theta Functions

Chapter 20 Theta Functions

…

20: 18.2 General Orthogonal Polynomials

… ►

Kernel Polynomials

… ► … ►

Sheffer Polynomials

… ►The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) …