Racah%20polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.2

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

ⓘ

Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

►is a polynomial of degree

n

, and hence a solution of (31.2.1) that is analytic at all three finite singularities

0,1,a

. These solutions are the Heun polynomials. …

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

… ►

Normalization

… ►

Orthogonal Invariance

… ►

Summation

… ►

Mean-Value

…

3: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

4: 18.3 Definitions

§18.3 Definitions

… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to

n=12

for these polynomials and for coefficients up to

n=6

for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►

Bessel polynomials

►Bessel polynomials are often included among the classical OP’s. …

5: 18.26 Wilson Class: Continued

… ►

§18.26(ii) Limit Relations

… ►

Racah $\to$ Dual Hahn

… ►

Racah $\to$ Hahn

… ►

Racah

… ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …

6: 18.25 Wilson Class: Definitions

… ►The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials

W_{n}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

. … ►

Further Constraints for Racah Polynomials

… ►

Racah

… ►Table 18.25.2 provides the leading coefficients

k_{n}

(§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …

7: 18.1 Notation

… ►

Classical OP’s

… ►

Hahn Class OP’s

… ►

Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$ .

… ►

$q$ -Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ .

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

8: 18.38 Mathematical Applications

… ►

Coding Theory

►For applications of Krawtchouk polynomials

K_{n}\left(x;p,N\right)

and

q

-Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)

to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). … ►The

6j

symbol (34.4.3), with an alternative expression as a terminating balanced

{{}_{4}F_{3}}

of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the

6j

symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … …

9: 18.28 Askey–Wilson Class

… ►Both the Askey–Wilson polynomials and the

q

-Racah polynomials can best be described as functions of

z

(resp. … ►

§18.28(viii) $q$ -Racah Polynomials

… ►These systems are the

q

-Racah polynomials and its limit cases. … ►

From $q$ -Racah to Racah

… ►Bannai and Ito (1984) introduced OP’s, called the Bannai–Ito polynomials which are the limit for

q\downarrow-1

of the

q

-Racah polynomials. …

10: 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

… ►

Meixner $\to$ Laguerre

…

Racah%20polynomials

1—10 of 316 matching pages

1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

Normalization

Orthogonal Invariance

Summation

Mean-Value

3: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

4: 18.3 Definitions

§18.3 Definitions

Bessel polynomials

5: 18.26 Wilson Class: Continued

§18.26(ii) Limit Relations

Racah → Dual Hahn

Racah → Hahn

Racah

6: 18.25 Wilson Class: Definitions

Further Constraints for Racah Polynomials

Racah

7: 18.1 Notation

Classical OP’s

Hahn Class OP’s

8: 18.38 Mathematical Applications

Coding Theory

9: 18.28 Askey–Wilson Class

§18.28(viii) q -Racah Polynomials

From q -Racah to Racah

10: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

§18.21(i) Dualities

§18.21(ii) Limit Relations and Special Cases

Hahn → Jacobi

Meixner → Laguerre

Racah $\to$ Dual Hahn

Racah $\to$ Hahn

§18.28(viii) $q$ -Racah Polynomials

From $q$ -Racah to Racah

Hahn $\to$ Jacobi

Meixner $\to$ Laguerre