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1: 18.40 Methods of Computation

… ►Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. … ►Convergence is

\sim O\left(N^{-2}\right)

. … ►Here

x(t,N)

is an interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

, that is,

x(i,N)=x_{i,N}

, allowing differentiation by

i

. …where the coefficients are defined recursively via

a_{1}=\frac{x_{1,N}}{x_{2,N}}-1

, and …The PWCF

x(t,N)

is a minimally oscillatory algebraic interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

. …

2: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with

N+1

regular singularities at

z=a_{j}

j=1,2,\dots,N

, and at

\infty

, is given by … ►

\alpha\beta=\sum_{j=1}^{N}a_{j}q_{j}.

►The three sets of parameters comprise the singularity parameters

a_{j}

, the exponent parameters

\alpha,\beta,\gamma_{j}

, and the

N-2

free accessory parameters

q_{j}

. With

a_{1}=0

and

a_{2}=1

the total number of free parameters is

3N-3

. Heun’s equation (31.2.1) corresponds to

N=3

. …

3: 19.19 Taylor and Related Series

… ►For

N=0,1,2,\dots

define the homogeneous hypergeometric polynomial …where the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

whose sum is

N

. … ►This form of

T_{N}

can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use …Then

T_{N}

has at most one term if

N\leq 5

in the series for

R_{F}

. For

R_{J}

and

R_{D}

T_{N}

has at most one term if

N\leq 3

, and two terms if

N=4

or 5. …

4: 32.14 Combinatorics

… ►Let

S_{N}

be the group of permutations

\boldsymbol{\pi}

of the numbers

1,2,\dots,N

(§26.2). With

1\leq m_{1}<\cdots<m_{n}\leq N

\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})

is said to be an increasing subsequence of

\boldsymbol{\pi}

of length

n

when

\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})

. Let

\ell_{N}(\boldsymbol{\pi})

be the length of the longest increasing subsequence of

\boldsymbol{\pi}

. … ►

32.14.1

\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),

ⓘ

Symbols:: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

…

5: 26.17 The Twelvefold Way

… ►The twelvefold way gives the number of mappings

f

from set

N

n

objects to set

K

k

objects (putting balls from set

N

into boxes in set

K

). … ►Table 26.17.1 is reproduced (in modified form) from Stanley (1997, p. 33). … ►

Table 26.17.1: The twelvefold way.

► ►►

elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
…

►

6: 31.15 Stieltjes Polynomials

… ►where

\Phi(z)

is a polynomial of degree not exceeding

N-2

. There exist at most

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

polynomials

V(z)

of degree not exceeding

N-2

such that for

\Phi(z)=V(z)

, (31.15.1) has a polynomial solution

w=S(z)

of degree

n

. … ►then there are exactly

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

polynomials

S(z)

, each of which corresponds to each of the

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

ways of distributing its

n

zeros among

N-1

intervals

(a_{j},a_{j+1})

j=1,2,\dots,N-1

. … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index

\mathbf{m}=(m_{1},m_{2},\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,\dots,N-1

. … ►Let

S_{\mathbf{m}}(z)

and

S_{\mathbf{l}}(z)

be Stieltjes polynomials corresponding to two distinct multi-indices

\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})

and

\mathbf{l}=(\ell_{1},\ell_{2},\dots,\ell_{N-1})

. …

7: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s

p_{n}(x)

with

x=\lambda(y)

: if the

y

-orthogonality set is

\{0,1,\dots,N\}

, then the role of the differentiation operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the operator

\Delta_{y}

followed by division by

\Delta_{y}(\lambda(y))

, or by the operator

\nabla_{y}

followed by division by

\nabla_{y}(\lambda(y))

. … ►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}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

. … ►If

\alpha+1=-N

, then the weights will be positive iff one of the following eight sets of inequalities holds: … ►

N-1<N+\gamma<\beta<-N-\delta.

… ►The first four sets imply

\gamma+\delta>-2

, and the last four imply

\gamma+\delta<-2N

. …

8: 3.6 Linear Difference Equations

… ►Stability can be restored, however, by backward recursion, provided that

c_{n}\neq 0

\forall n

: starting from

w_{N}

and

w_{N+1}

, with

N

large, equation (3.6.3) is applied to generate in succession

w_{N-1},w_{N-2},\dots,w_{0}

. … ►Here

N

is an arbitrary positive integer. … ►starting with

w_{N}=0

. … ►The least value of

N

that satisfies (3.6.9) is found to be 16. …

9: 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

… ►

18.23.3

\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\genfrac{(}{)}{0.0% pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},

x=0,1,\dots,N

ⓘ

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

…

10: 18.26 Wilson Class: Continued

… ►Here we use as convention for (16.2.1) with

b_{q}=-N

a_{1}=-n

, and

n=0,1,\ldots,N

that the summation on the right-hand side ends at

k=n

. … ►

18.26.11

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

ⓘ

Symbols:: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : 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

… ►

18.26.12

r(x;\beta,c,N)=x(x+\beta+c^{-1}(1-c)N),

ⓘ

Defines:: $r(x;\beta,c,N)$ (locally)
Symbols:: $N$ : positive integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

►

18.26.13

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

ⓘ

Symbols:: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : 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

… ►

18.26.21

(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}% ^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N% \right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

…