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1: 18.21 Hahn Class: Interrelations

… ►

$K_n(x;p,N)=K_x(n;p,N),$ $n,x=0,1,\mathrm{\dots },N$.

… ►

18.21.3 $$\underset{t\to \mathrm{\infty }}{lim}{Q}_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $K_n(x;p,N)$: 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

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18.21.4 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(x;\beta -1,N(c^{-1}-1),N)=M_n(x;\beta ,c).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $M_n(x;\beta ,c)$: 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 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(Nx;\alpha ,\beta ,N)=\frac{P_n^{(\alpha ,\beta )}\left(1-2x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}.$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: 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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18.21.6 $$\underset{N\to \mathrm{\infty }}{lim}{K}_n(x;N^{-1}a,N)=C_n(x;a).$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E6

Encodings:

pMML, png, TeX

See also:

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

…

2: 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 $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. ►

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	$Q_n(x;\alpha ,\beta ,N)$, $n=0,1,\mathrm{\dots },N$	$\{0,1,\mathrm{\dots },N\}$	${\displaystyle \frac{{\left(\alpha +1\right)}_x{\left(\beta +1\right)}_{N-x}}{x!(N-x)!}}$, $\alpha ,\beta >-1$ or	${\displaystyle \frac{{(-1)}^n{\left(n+\alpha +\beta +1\right)}_{N+1}{\left(\beta +1\right)}_{n}n!}{(2n+\alpha +\beta +1){\left(\alpha +1\right)}_n{\left(-N\right)}_{n}N!}}$ If , then ${(-1)}^N{w}_x>0$ and ${(-1)}^N{h}_n>0$.
Krawtchouk	$K_n(x;p,N)$, $n=0,1,\mathrm{\dots },N$	$\{0,1,\mathrm{\dots },N\}$	$\left({\displaystyle \genfrac{}{}{0pt}{}{N}{x}}\right)p^x{(1-p)}^{N-x}$,	${\left({\displaystyle \frac{1-p}{p}}\right)}^n/\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)$
…

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►

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

► ►►►►

$p_n(x)$	$k_n$
$Q_n(x;\alpha ,\beta ,N)$	${\displaystyle \frac{{\left(n+\alpha +\beta +1\right)}_n}{{\left(\alpha +1\right)}_n{\left(-N\right)}_n}}$
$K_n(x;p,N)$	$p^{-n}/{\left(-N\right)}_n$
…

►

…

3: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.1 $$p_n(x)=Q_n(x;\alpha ,\beta ,N),$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: 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

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18.22.9 $$p_n(x)=Q_n(x;\alpha ,\beta ,N),$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: 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 $${\mathrm{\Delta }}_x{Q}_n(x;\alpha ,\beta ,N)=-\frac{n(n+\alpha +\beta +1)}{(\alpha +1)N}{Q}_{n-1}(x;\alpha +1,\beta +1,N-1),$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $\mathrm{\Delta }$: 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

… ►

18.22.21 $${\mathrm{\Delta }}_x{K}_n(x;p,N)=-\frac{n}{pN}{K}_{n-1}(x;p,N-1),$$

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

►

18.22.22 $${\nabla }_x\left(\left(\genfrac{}{}{0.0pt}{}{N}{x}\right)p^x{(1-p)}^{N-x}{K}_n(x;p,N)\right)=\left(\genfrac{}{}{0.0pt}{}{N+1}{x}\right)p^x{(1-p)}^{N-x}{K}_{n+1}(x;p,N+1).$$

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, ${\nabla }_x$: backward difference, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

…

4: 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) §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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18.26.9 $$\underset{\beta \to \mathrm{\infty }}{lim}{R}_n(x;-N-1,\beta ,\gamma ,\delta )=R_n(x;\gamma ,\delta ,N).$$

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn 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.E9

Encodings:

pMML, png, TeX

See also:

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

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18.26.10 $$\underset{\delta \to \mathrm{\infty }}{lim}{R}_n(x(x+\gamma +\delta +1);\alpha ,\beta ,-N-1,\delta )=Q_n(x;\alpha ,\beta ,N).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta )$: 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

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

ⓘ

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

ⓘ

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

…

5: 18.20 Hahn Class: Explicit Representations

… ►For the Hahn polynomials $p_n(x)=Q_n(x;\alpha ,\beta ,N)$ and … ►

Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

► ►►►

$p_n(x)$	$F(x)$	${\kappa }_n$
$K_n(x;p,N)$	$x-N$	${\left(-N\right)}_n$
…

►

… ►

18.20.5 $$Q_n(x;\alpha ,\beta ,N)={}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +\beta +1,-x}{\alpha +1,-N};1),$$ $n=0,1,\mathrm{\dots },N$.

ⓘ

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, $Q_n(x;\alpha ,\beta ,N)$: 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

►

18.20.6 $$K_n(x;p,N)={}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,-x}{-N};p^{-1}),$$ $n=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.21(ii)

Permalink:

http://dlmf.nist.gov/18.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

…

6: 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.

► ►►►

OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
…
dual Hahn	$R_n(x;\gamma ,\delta ,N)$	$y(y+\gamma +\delta +1)$	$\{0,1,\mathrm{\dots },N\}$	$\gamma ,\delta >-1$ or

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… ►

18.25.10 $$p_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta ),$$ $\alpha +1=-N$,

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E10

Encodings:

pMML, png, TeX

See also:

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

… ►

18.25.13 $$p_n(x)=R_n(x;\gamma ,\delta ,N),$$

ⓘ

Symbols:

$R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E13

Encodings:

pMML, png, TeX

See also:

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

… ►

18.25.15 $$h_n=\frac{n!(N-n)!{\left(\gamma +\delta +2\right)}_N}{N!{\left(\gamma +1\right)}_n{\left(\delta +1\right)}_{N-n}}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E15

Encodings:

pMML, png, TeX

See also:

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

…

7: 18.23 Hahn Class: Generating Functions

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18.23.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-x}{\alpha +1};-z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{x-N}{\beta +1};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n}{{\left(\beta +1\right)}_{n}n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: 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 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

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, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: 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

… ►

18.23.3 $${\left(1-\frac{1-p}{p}z\right)}^x{(1+z)}^{N-x}=\sum _{n=0}^N\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)K_n(x;p,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

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

…

8: 19.19 Taylor and Related Series

… ►For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial ►

19.19.1 $$T_N(𝐛,𝐳)=\sum \frac{{\left(b_1\right)}_{m_1}\mathrm{\cdots }{\left(b_n\right)}_{m_n}}{m_1!\mathrm{\cdots }{m}_n!}{z}_1^{m_1}\mathrm{\cdots }{z}_n^{m_n},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.20(v)

Permalink:

http://dlmf.nist.gov/19.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

… ►

19.19.2 $$R_{-a}(𝐛;𝐳)=\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(𝐛,\mathrm{𝟏}-𝐳),$$ $c={\sum }_{j=1}^n{b}_j$, ,

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

… ►

19.19.5 $$T_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐳)=\sum {(-1)}^{M+N}{\left(\frac{1}{2}\right)}_M\frac{E_1^{m_1}(𝐳)\mathrm{\cdots }{E}_n^{m_n}(𝐳)}{m_1!\mathrm{\cdots }{m}_n!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $E_s(𝐳)$: symmetric function and $M$: sum

Referenced by:

§19.19, §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

… ►

19.19.7 $$R_{-a}(\frac{\mathrm{𝟏}}{\mathrm{𝟐}};𝐳)=A^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(\frac{1}{2}n\right)}_N}{T}_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐙),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $𝐙$: variable and $A$

Referenced by:

§19.15, §19.36(i), §19.36(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

…

9: 18.1 Notation

… ►

Hahn: $Q_n(x;\alpha ,\beta ,N)$.

►

Krawtchouk: $K_n(x;p,N)$.

… ►

$q$-Hahn: $Q_n(x;\alpha ,\beta ,N;q)$.

…

10: 18.3 Definitions

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: ►

18.3.1 $$\sum _{n=1}^{N+1}{T}_j\left(x_{N+1,n}\right)T_k\left(x_{N+1,n}\right)=0,$$ $0\le j\le N$, $0\le k\le N$, $j\ne k$,

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.3, §18.3

Permalink:

http://dlmf.nist.gov/18.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.3, §18.3 and Ch.18

… ►For $\nu \in ℝ$ and $N>-\frac{1}{2}$ a finite system of Jacobi polynomials $P_n^{(-N-1+\mathrm{i}\nu ,-N-1-\mathrm{i}\nu )}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$ with $w(x)={\left(1+x^2\right)}^{-N-1}{\mathrm{e}}^{2\nu \arctan x}$. …