dual Hahn polynomials

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1: 18.25 Wilson Class: Definitions

… ►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 $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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continuous dual Hahn	$S_n(x;a,b,c)$	$y^2$	$(0,\mathrm{\infty })$	$\mathrm{\Re }(a,b,c)>0$; nonreal parameters in conjugate pairs
…

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►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\mathrm{\infty })\cup S$, where $S$ is a specific finite set, e. … ►

18.25.6 $$p_n(x)=S_n(x;a_1,a_2,a_3),$$

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Symbols:

$S_n(x;a,b,c)$: continuous dual Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.25(ii)

Permalink:

http://dlmf.nist.gov/18.25.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

… ►Table 18.25.2 provides the leading coefficients $k_n$ (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …

2: 18.26 Wilson Class: Continued

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18.26.4_1 $$R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)=Q_y(n;\gamma ,\delta ,N),$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

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18.26.5 $$\underset{d\to \mathrm{\infty }}{lim}\frac{W_n(x;a,b,c,d)}{{\left(a+d\right)}_n}=S_n(x;a,b,c).$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_n(x;a,b,c,d)$: Wilson polynomial, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.8 $$\underset{t\to \mathrm{\infty }}{lim}{S}_n({(x-t)}^2;\lambda +it,\lambda -it,t\cot \varphi )/t^n=n!{(\csc \varphi )}^n{P}_n^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $\csc z$: cosecant function, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.11 $$\underset{t\to \mathrm{\infty }}{lim}{R}_n(x(x+t+1);pt,(1-p)t,N)=K_n(x;p,N).$$

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.13 $$\underset{N\to \mathrm{\infty }}{lim}{R}_n(r(x;\beta ,c,N);\beta -1,c^{-1}(1-c)N,N)=M_n(x;\beta ,c).$$

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $n$: nonnegative integer, $x$: real variable and $r(x;\beta ,c,N)$

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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3: 18.21 Hahn Class: Interrelations

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18.21.1 $$Q_n(x;\alpha ,\beta ,N)=R_x(n(n+\alpha +\beta +1);\alpha ,\beta ,N),$$ $n,x=0,1,\mathrm{\dots },N$.

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.26(i), §18.38(iii)

Permalink:

http://dlmf.nist.gov/18.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(i), §18.21(i), §18.21 and Ch.18

►For the dual Hahn polynomial $R_n(x;\gamma ,\delta ,N)$ see §18.25. …

4: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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5: 18.38 Mathematical Applications

… ►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. … …