q-Hahn class orthogonal polynomials

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11: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s

p_{n}(x)

with

x=\lambda(y)

: if the

y

-orthogonality set is

\{0,1,\dots,N\}

, then the role of the differentiation operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the operator

\Delta_{y}

followed by division by

\Delta_{y}(\lambda(y))

, or by the operator

\nabla_{y}

followed by division by

\nabla_{y}(\lambda(y))

. …The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). … ►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is

(0,\infty)\cup S

, where

S

is a specific finite set, e. … ►

§18.25(ii) Weights and Standardizations: Continuous Cases

…

12: 18.38 Mathematical Applications

… ►

Quadrature

… ►

Riemann–Hilbert Problems

… ►

Group Representations

… ► … ► …

13: 18.36 Miscellaneous Polynomials

… ►

§18.36(ii) Sobolev Orthogonal Polynomials

… ►

§18.36(iii) Multiple Orthogonal Polynomials

… ►

§18.36(iv) Orthogonal Matrix Polynomials

… ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►

§18.36(vi) Exceptional Orthogonal Polynomials

…

14: 18.27 $q$ -Hahn Class

§18.27 $q$ -Hahn Class

… ►All these systems of OP’s have orthogonality properties of the form … ►

From Big $q$ -Jacobi to Jacobi

… ►

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

… ►

Limit Relations

…

15: 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 …For the Krawtchouk, Meixner, and Charlier polynomials,

F(x)

and

\kappa_{n}

are as in Table 18.20.1. … ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

…

16: 18.26 Wilson Class: Continued

… ►

§18.26(ii) Limit Relations

… ►See also Figure 18.21.1. ►

§18.26(iii) Difference Relations

… ►

§18.26(iv) Generating Functions

… ►

§18.26(v) Asymptotic Approximations

…

17: 18.2 General Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

… ►

§18.2(xi) Some Special Classes of General Orthogonal Polynomials

►

The Szegő Class $\mathcal{G}$

… ► … ►In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) …

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. …

19: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

… ►

Hahn

… ►

18.23.3

\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\genfrac{(}{)}{0.0% pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},

x=0,1,\dots,N

ⓘ

Symbols:: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

18.23.5

{\mathrm{e}}^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}% \left(x;a\right)}{n!}z^{n},

x=0,1,2,\dots

ⓘ

Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.2(xii), §18.23
Permalink:: http://dlmf.nist.gov/18.23.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.7

(1-{\mathrm{e}}^{\mathrm{i}\phi}z)^{-\lambda+\mathrm{i}x}(1-{\mathrm{e}}^{-% \mathrm{i}\phi}z)^{-\lambda-\mathrm{i}x}=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}% \left(x;\phi\right)z^{n},

|z|<1

ⓘ

Symbols:: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : 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

20: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

… ► … ►

§18.28(ii) Askey–Wilson Polynomials

… ►

§18.28(x) Limit Relations

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).