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英语四级会发成绩单吗【仿证微CXFK69】hahnq

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1: 18.21 Hahn Class: Interrelations

… ►

18.21.1

Q_{n}\left(x;\alpha,\beta,N\right)=R_{x}\left(n(n+\alpha+\beta+1);\alpha,\beta% ,N\right),

n,x=0,1,\dots,N

.

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : 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

… ►

18.21.3

\lim_{t\to\infty}Q_{n}\left(x;pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $N$ : positive integer, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.4

\lim_{N\to\infty}Q_{n}\left(x;\beta-1,N(c^{-1}-1),N\right)=M_{n}\left(x;\beta,% c\right).

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.21.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.5

\lim_{N\to\infty}Q_{n}\left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_% {n}\left(1-2x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}.

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.21.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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2: 18.23 Hahn Class: Generating Functions

… ►

18.23.1

{{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

.

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

►

18.23.2

{{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

…

3: 18.19 Hahn Class: Definitions

… ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials

Q_{n}\left(x;\alpha,\beta,N\right)

, Krawtchouk polynomials

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

, Meixner polynomials

M_{n}\left(x;\beta,c\right)

, and Charlier polynomials

C_{n}\left(x;a\right)

. ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

	$p_{n}(x)$	$X$	$w_{x}$	$h_{n}$
Hahn	$Q_{n}\left(x;\alpha,\beta,N\right)$ , $n=0,1,\ldots,N$	$\{0,1,\ldots,N\}$	$\dfrac{{\left(\alpha+1\right)_{x}}{\left(\beta+1\right)_{N-x}}}{x!(N-x)!}$ , $\alpha,\beta>-1$ or $\alpha,\beta<-N$	$\dfrac{(-1)^{n}{\left(n+\alpha+\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}% n!}{(2n+\alpha+\beta+1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}$ If $\alpha,\beta<-N$ , then $(-1)^{N}w_{x}>0$ and $(-1)^{N}h_{n}>0$ .
…

► ►

Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

$p_{n}(x)$	$k_{n}$
$Q_{n}\left(x;\alpha,\beta,N\right)$	$\dfrac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(% -N\right)_{n}}}$
…

► …

4: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.1

p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

… ►

18.22.9

p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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18.22.19

\Delta_{x}Q_{n}\left(x;\alpha,\beta,N\right)=-\frac{n(n+\alpha+\beta+1)}{(% \alpha+1)N}Q_{n-1}\left(x;\alpha+1,\beta+1,N-1\right),

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $\Delta$ : forward difference operator, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §16.4(iii), §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

►

18.22.20

\nabla_{x}\left(\frac{{\left(\alpha+1\right)_{x}}{\left(\beta+1\right)_{N-x}}}% {x!\;(N-x)!}Q_{n}\left(x;\alpha,\beta,N\right)\right)=\frac{N+1}{\beta}\frac{{% \left(\alpha\right)_{x}}{\left(\beta\right)_{N+1-x}}}{x!\;(N+1-x)!}\*Q_{n+1}% \left(x;\alpha-1,\beta-1,N+1\right).

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\nabla_{\NVar{x}}$ : backward difference, $!$ : factorial (as in $n!$ ), $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/18.22.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

…

5: 18.20 Hahn Class: Explicit Representations

… ►For the Hahn polynomials

p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right)

and … ►

18.20.5

Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),

n=0,1,\dots,N

.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.20(ii), §18.21(ii), §18.23, §18.26(ii), §18.27(ii), §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

…

6: 18.26 Wilson Class: Continued

… ►

18.26.4_1

R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)=Q_{y}\left(n;\gamma,% \delta,N\right),

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : 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

… ►

18.26.10

\lim_{\delta\to\infty}R_{n}\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta% \right)=Q_{n}\left(x;\alpha,\beta,N\right).

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

…

7: 18.1 Notation

… ►

Hahn: $Q_{n}\left(x;\alpha,\beta,N\right)$ .

…

8: 18.27 $q$ -Hahn Class

… ►

18.27.4_2

\lim_{q\to 1}Q_{n}\left(q^{-x};q^{\alpha},q^{\beta},N;q\right)=Q_{n}\left(x;% \alpha,\beta,N\right).

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N};\NVar{q}\right)$ : $q$ -Hahn polynomial, $N$ : positive integer, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.27(ii), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.27.E4_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.27(ii), §18.27 and Ch.18

…

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