continuous 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:

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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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18.26.2 $$S_n(y^2;a,b,c)={\left(a+b\right)}_n{\left(a+c\right)}_n{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a+iy,a-iy}{a+b,a+c};1).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E2

Encodings:

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Annotations for §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:

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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:

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See also:

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

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18.26.15 $${\delta }_y\left(S_n(y^2;a,b,c)\right)/{\delta }_y(y^2)=-n{S}_{n-1}(y^2;a+\frac{1}{2},b+\frac{1}{2},c+\frac{1}{2}).$$

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Symbols:

${\delta }_x$: central difference, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.26.E15

Encodings:

pMML, png, TeX

See also:

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

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18.26.19 $${(1-z)}^{-c+\mathrm{i}y}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}y,b+\mathrm{i}y}{a+b};z)=\sum _{n=0}^{\mathrm{\infty }}\frac{S_n(y^2;a,b,c)}{{\left(a+b\right)}_{n}n!}{z}^n,$$ .

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $y$: real variable, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E19

Encodings:

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See also:

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

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3: 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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4: 18.38 Mathematical Applications

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Approximation Theory

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Integrable Systems

… ►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. … …