Meixner polynomials

(0.002 seconds)

11—20 of 25 matching pages

11: 15.9 Relations to Other Functions

… ►

Meixner

►

15.9.9

M_{n}\left(x;\beta,c\right)=F\left({-n,-x\atop\beta};1-\frac{1}{c}\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $x$ : real variable, $n$ : integer and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

►

Meixner–Pollaczek

►

15.9.10

P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{% \mathrm{e}}^{n\mathrm{i}\phi}F\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-{% \mathrm{e}}^{-2\mathrm{i}\phi}\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $x$ : real variable and $n$ : integer
Permalink:: http://dlmf.nist.gov/15.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

…

12: Bibliography J

… ►

X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

►

X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

…

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. … ►

Equation (18.35.9)

18.35.9

P^{(\lambda)}_{n}\left(x;\phi\right)=P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin% \phi\right),

P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=P^{(\lambda)}_{n}\left(\tau_{a,b}% (\theta);\theta\right)

Previously we gave only the first identity $P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin\phi\right)=P^{(\lambda)}_{n}\left(x;% \phi\right)$ .

…

14: Bibliography W

… ►

X.-S. Wang and R. Wong (2011) Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

ⓘ

External Links:: ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt
Referenced by:: §18.24, §18.24.
Encodings:: BibTeX

…

15: Bibliography K

… ►

S. F. Khwaja and A. B. Olde Daalhuis (2012) Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.

ⓘ

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

… ►

T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §18.21(ii).
Encodings:: BibTeX

…

16: Bibliography L

… ►

X. Li and R. Wong (2001) On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

…

17: 18.2 General Orthogonal Polynomials

… ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … ►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). …

18: Bibliography B

… ►

P. L. Butzer and T. H. Koornwinder (2019) Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

ⓘ

External Links:: ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

…

19: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. … Meixner and F. …

20: 28.34 Methods of Computation

… ►

(e)

Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

►

(f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

… ►

(d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

…