Hahn polynomials

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1: 18.21 Hahn Class: Interrelations

… ►

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

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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. … ►

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

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

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

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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

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

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18.21.5 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(Nx;\alpha ,\beta ,N)=\frac{P_n^{(\alpha ,\beta )}\left(1-2x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}.$$

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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

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

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2: 18.19 Hahn Class: Definitions

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

► ►►►

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

►

… ► ►

18.19.1 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

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Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E1

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Annotations for §18.19, §18.19 and Ch.18

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3: 18.22 Hahn Class: Recurrence Relations and Differences

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

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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:

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See also:

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

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18.22.4 $$q_n(x)=p_n(x;a,b,\overline{a},\overline{b})/p_n(\mathrm{i}a;a,b,\overline{a},\overline{b}),$$

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Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(i)

Permalink:

http://dlmf.nist.gov/18.22.E4

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

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

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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

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See also:

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

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18.22.13 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

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Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(ii)

Permalink:

http://dlmf.nist.gov/18.22.E13

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See also:

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

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18.22.19 $${\mathrm{\Delta }}_x{Q}_n(x;\alpha ,\beta ,N)=-\frac{n(n+\alpha +\beta +1)}{(\alpha +1)N}{Q}_{n-1}(x;\alpha +1,\beta +1,N-1),$$

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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

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

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4: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

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Hahn

►

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

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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:

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See also:

Annotations for §18.23, §18.23 and Ch.18

►

18.23.2 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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

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Annotations for §18.23, §18.23 and Ch.18

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18.23.6 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}x}{2\mathrm{\Re }a};-\mathrm{i}z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\overline{b}-\mathrm{i}x}{2\mathrm{\Re }b};\mathrm{i}z)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x;a,b,\overline{a},\overline{b})}{{\left(2\mathrm{\Re }a\right)}_n{\left(2\mathrm{\Re }b\right)}_n}{z}^n.$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E6

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Annotations for §18.23, §18.23 and Ch.18

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5: 18.20 Hahn Class: Explicit Representations

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

18.20.3 $$w(x;a,b,\overline{a},\overline{b})p_n(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta }_x^n\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E3

Encodings:

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See also:

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

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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

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

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

… ►(For symmetry properties of $p_n(x;a,b,\overline{a},\overline{b})$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …

6: 37.10 Other Orthogonal Polynomials of Two Variables

… ►

§37.10(iv) Hahn polynomials of Two Variables

… ►As an example we give the Hahn polynomials of two variables: ►

37.10.5 $$Q_{k,n}(x,y;\alpha ,\beta ,\gamma ,N)=Q_{n-k}(x;\alpha ,\beta +\gamma +2k+1,N-k)\frac{{\left(-N+x\right)}_k}{{\left(-N\right)}_k}{Q}_k(y;\beta ,\gamma ,N-x),$$ $0\le k\le n\le N$,

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $n$: nonnegative integer, $Q$: polynomial of two variables, $x$: real variable and $y$: real variable

Source:

Rahman (1981b, (3.10))

Referenced by:

§37.10(iv)

Permalink:

http://dlmf.nist.gov/37.10.E5

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Annotations for §37.10(iv), §37.10, Part 37.II and Ch.37

►where the Hahn polynomial $Q_k(x;\alpha ,\beta ,N)$ is given by (18.20.5). … ►

37.10.7 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_{k,n}(Nx,Ny;\alpha ,\beta ,\gamma ,N)=\frac{P_{k,n}^{\alpha ,\beta ,\gamma }(x,y)}{P_{k,n}^{\alpha ,\beta ,\gamma }(0,0)}.$$

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Symbols:

$k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $Q$: polynomial of two variables, $x$: real variable and $y$: real variable

Proof sketch:

Use (18.20.5), (18.5.7), the symmetry of Jacobi polynomials given in Table 18.6.1, and (37.3.3).

Permalink:

http://dlmf.nist.gov/37.10.E7

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Annotations for §37.10(iv), §37.10, Part 37.II and Ch.37

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7: 18.1 Notation

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

… ►

Hahn Class OP’s

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

… ►

Continuous Hahn: $p_n(x;a,b,\overline{a},\overline{b})$.

… ►

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

…

8: 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

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

ⓘ

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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18.26.6 $$\underset{t\to \mathrm{\infty }}{lim}\frac{W_n({(x+t)}^2;a-it,b-it,\overline{a}+it,\overline{b}+it)}{{(-2t)}^{n}n!}=p_n(x;a,b,\overline{a},\overline{b}).$$

ⓘ

Symbols:

$W_n(x;a,b,c,d)$: Wilson polynomial, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(i), §18.22(ii), §18.23, §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E6

Encodings:

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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:

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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:

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

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9: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

10: 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
…
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
…

►

►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 $$p_n(x)=S_n(x;a_1,a_2,a_3),$$

ⓘ

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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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. …