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11: 33.26 Software

… ►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web. … ►

•

Noble (2004). Fortran 90.

…

12: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that …Here

\xi(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. For general

\boldsymbol{{\Gamma}}

, it is difficult to decide which root needs to be used. … ►(

\mathbf{A}

invertible with integer elements.) …For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. …

13: 26.8 Set Partitions: Stirling Numbers

… ►

Table 26.8.1: Stirling numbers of the first kind

s\left(n,k\right)

► ►►►

$n$	$k$
$n$	…
6	$0$	$-120$	$274$	$-225$	$85$	$-15$	$1$
…

► ►

Table 26.8.2: Stirling numbers of the second kind

S\left(n,k\right)

► ►►►►

$n$	$k$
$n$	…
6	0	1	31	90	65	15	1
…
8	0	1	127	966	1701	1050	266	28	1
…

► … ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. … ►

26.8.38

A^{-1}=B.

ⓘ

A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E38
Encodings:: pMML, png, TeX
See also:: Annotations for §26.8(v), §26.8 and Ch.26

…

14: 16.25 Methods of Computation

… ►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).

15: 1.11 Zeros of Polynomials

… ►Set

z=w-\tfrac{1}{3}a

to reduce

f(z)=z^{3}+az^{2}+bz+c

g(w)=w^{3}+pw+q

, with

p=(3b-a^{2})/3

q=(2a^{3}-9ab+27c)/27

. … ►

f(z)=z^{3}-6z^{2}+6z-2

g(w)=w^{3}-6w-6

A=3\sqrt[3]{4}

B=3\sqrt[3]{2}

. … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►Let … ►Then

f(z)

, with

a_{n}\not=0

, is stable iff

a_{0}\not=0

;

D_{2k}>0

k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor

;

\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}

k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor

16: 26.13 Permutations: Cycle Notation

… ►An explicit representation of

\sigma

can be given by the

2\times n

matrix: … ►is

{\left(1,3,2,5,7\right)}{\left(4\right)}{\left(6,8\right)}

in cycle notation. …In consequence, (26.13.2) can also be written as

{\left(1,3,2,5,7\right)}{\left(6,8\right)}

. … ►A permutation that consists of a single cycle of length

k

can be written as the composition of

k-1

two-cycles (read from right to left): …A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

17: 20.15 Tables

… ►This reference gives

\theta_{j}\left(x,q\right)

j=1,2,3,4

, and their logarithmic

x

-derivatives to 4D for

x/\pi=0(.1)1

\alpha=0(9^{\circ})90^{\circ}

, where

\alpha

is the modular angle given by … ►Spenceley and Spenceley (1947) tabulates

\theta_{1}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{2}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{3}\left(x,q\right)/\theta_{4}\left(0,q\right)

\theta_{4}\left(x,q\right)/\theta_{4}\left(0,q\right)

to 12D for

u=0(1^{\circ})90^{\circ}

\alpha=0(1^{\circ})89^{\circ}

, where

u=2x/(\pi{\theta_{3}}^{2}\left(0,q\right))

and

\alpha

is defined by (20.15.1), together with the corresponding values of

\theta_{2}\left(0,q\right)

and

\theta_{4}\left(0,q\right)

. ►Lawden (1989, pp. 270–279) tabulates

\theta_{j}\left(x,q\right)

j=1,2,3,4

, to 5D for

x=0(1^{\circ})90^{\circ}

q=0.1(.1)0.9

, and also

q

to 5D for

k^{2}=0(.01)1

. ►Tables of Neville’s theta functions

\theta_{s}\left(x,q\right)

\theta_{c}\left(x,q\right)

\theta_{d}\left(x,q\right)

\theta_{n}\left(x,q\right)

(see §20.1) and their logarithmic

x

-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for

\varepsilon,\alpha=0(5^{\circ})90^{\circ}

, where (in radian measure)

\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))

, and

\alpha

is defined by (20.15.1). …

18: 8.23 Statistical Applications

… ►The function

\mathrm{B}_{x}\left(a,b\right)

and its normalization

I_{x}\left(a,b\right)

play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of

Q\left(a,x\right)

; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

19: 11.9 Lommel Functions

… ►can be regarded as a generalization of (11.2.7). …where

A

B

are arbitrary constants,

s_{{\mu},{\nu}}\left(z\right)

is the Lommel function defined by … ►

11.9.4

a_{k}(\mu,\nu)=\prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)=4^{k}{\left(% \frac{\mu-\nu+1}{2}\right)_{k}}{\left(\frac{\mu+\nu+1}{2}\right)_{k}},

k=0,1,2,\dots

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\nu$ : real or complex order, $k$ : nonnegative integer and $a_{k}(\mu,\nu)$ : expansion function
Referenced by:: §11.9(iii), Erratum (V1.1.3) for Additions
Permalink:: http://dlmf.nist.gov/11.9.E4
Encodings:: pMML, png, TeX
Addition (effective with 1.1.3):: An alternative Pochhammer symbol representation was added.
See also:: Annotations for §11.9(i), §11.9 and Ch.11

… ►the right-hand side being replaced by its limiting form when

\mu\pm\nu

is an odd negative integer. … ►For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32). …

20: 5.16 Sums

… ►

5.16.1

\sum_{k=1}^{\infty}(-1)^{k}\psi'\left(k\right)=-\frac{\pi^{2}}{8},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.16 and Ch.5

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).