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11: 18.19 Hahn Class: Definitions

… ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§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, normalizations, and parameter constraints.

► ►►►

$p_n(x)$	$X$	$w_x$	$h_n$
…
$C_n(x;a)$	$\{0,1,2,\mathrm{\dots }\}$	$a^x/x!$, $a>0$	$a^{-n}{\mathrm{e}}^{a}n!$

►

… ►These polynomials are orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$, and with $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$ are defined as follows. ►

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

ⓘ

Symbols:

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

Referenced by:

§18.19

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

…

12: Roelof Koekoek

… ►Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. …

13: 29.17 Other Solutions

… ►If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by …

14: 33.14 Definitions and Basic Properties

… ►When $ϵ=-1/n^2$, $n=\mathrm{\ell }+1,\mathrm{\ell }+2,\mathrm{\dots }$, $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ times a polynomial in $r/n$, and …

15: 13.6 Relations to Other Functions

… ►

13.6.16 $$M(-n,\frac{1}{2},z^2)={(-1)}^n\frac{n!}{(2n)!}{H}_{2n}\left(z\right),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.17

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

►

13.6.17 $$M(-n,\frac{3}{2},z^2)={(-1)}^n\frac{n!}{(2n+1)!2z}{H}_{2n+1}\left(z\right),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.18 (as corrected in later editions)

Permalink:

http://dlmf.nist.gov/13.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

►

13.6.18 $$U(\frac{1}{2}-\frac{1}{2}n,\frac{3}{2},z^2)=2^{-n}{z}^{-1}{H}_n\left(z\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.38 (as corrected in later editions)

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E18

Encodings:

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See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

… ►

13.6.19 $$U(-n,\alpha +1,z)={(-1)}^n{\left(\alpha +1\right)}_{n}M(-n,\alpha +1,z)={(-1)}^{n}n!L_n^{(\alpha )}\left(z\right).$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.9 and 13.6.27

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

… ►

13.6.20 $$U(-n,z-n+1,a)={\left(-z\right)}_{n}M(-n,z-n+1,a)=a^n{C}_n(z;a).$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

…

16: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. …

17: 31.8 Solutions via Quadratures

… ►

31.8.2 $$w_{\pm }(\mathbf{m};\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

ⓘ

Symbols:

$dx$: differential, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

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See also:

Annotations for §31.8 and Ch.31

… ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …

18: 31.15 Stieltjes Polynomials

… ►where $\mathrm{\Phi }(z)$ is a polynomial of degree not exceeding $N-2$. There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi }(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree $n$. … ►If $z_1,z_2,\mathrm{\dots },z_n$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero $z_k$ is either one of the parameters $a_j$ or a solution of the system of equations … ►If $t_k$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_1^{\prime },z_2^{\prime },\mathrm{\dots },z_{n-1}^{\prime }$ are the zeros of $S^{\prime }(z)$ (the derivative of $S(z)$), then $t_k$ is either a zero of $S^{\prime }(z)$ or a solution of the equation … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $\mathbf{m}=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …

19: 18.22 Hahn Class: Recurrence Relations and Differences

… ►These polynomials satisfy (18.22.2) with $p_n(x)$, $A_n$, and $C_n$ as in Table 18.22.1. … ►

18.22.20 $${\nabla }_x\left(\frac{{\left(\alpha +1\right)}_x{\left(\beta +1\right)}_{N-x}}{x!(N-x)!}{Q}_n(x;\alpha ,\beta ,N)\right)=\frac{N+1}{\beta }\frac{{\left(\alpha \right)}_x{\left(\beta \right)}_{N+1-x}}{x!(N+1-x)!}{Q}_{n+1}(x;\alpha -1,\beta -1,N+1).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer, $N$: positive 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

… ►

18.22.24 $${\nabla }_x\left(\frac{{\left(\beta \right)}_x{c}^x}{x!}{M}_n(x;\beta ,c)\right)=\frac{{\left(\beta -1\right)}_x{c}^x}{x!}{M}_{n+1}(x;\beta -1,c).$$

ⓘ

Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E24

Encodings:

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See also:

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

… ►

18.22.25 $${\mathrm{\Delta }}_x{C}_n(x;a)=-\frac{n}{a}{C}_{n-1}(x;a),$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E25

Encodings:

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See also:

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

►

18.22.26 $${\nabla }_x\left(\frac{a^x}{x!}{C}_n(x;a)\right)=\frac{a^x}{x!}{C}_{n+1}(x;a).$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E26

Encodings:

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See also:

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

…

20: 18.35 Pollaczek Polynomials

… ►

$P_0^{(\lambda )}(x;a,b)=1,$

… ►

18.35.2 $$(n+1)P_{n+1}^{(\lambda )}(x;a,b)=2((n+\lambda +a)x+b)P_n^{(\lambda )}(x;a,b)-(n+2\lambda -1)P_{n-1}^{(\lambda )}(x;a,b),$$ $n=0,1,\mathrm{\dots }$.

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.35.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(i), §18.35 and Ch.18

… ►

18.35.5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b)P_m^{(\lambda )}(x;a,b)w^{(\lambda )}(x;a,b)dx=0,$$ $n\ne m$,

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $dx$: differential, $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.35.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

… ►See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, with $a$ and $b$ fixed. …Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$.