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21: 18.38 Mathematical Applications

… ►For the generalized hypergeometric function

{{}_{3}F_{2}}

see (16.2.1). … ►See, for example, Andrews et al. (1999, Chapter 9). … ►The

3j

symbol (34.2.6), with an alternative expression as a terminating

{{}_{3}F_{2}}

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The

6j

symbol (34.4.3), with an alternative expression as a terminating balanced

{{}_{4}F_{3}}

of unit argument, can be expressend in terms of Racah polynomials (18.26.3). … ►The abstract associative algebra with generators

K_{0},K_{1},K_{2}

and relations (18.38.4), (18.38.6) and with the constants

B,C_{0},C_{1},D_{0},D_{1}

in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). …

22: 3.5 Quadrature

… ►If in addition

f

is periodic,

f\in C^{k}(\mathbb{R})

, and the integral is taken over a period, then … ►If

f\in C^{2m+2}[a,b]

, then the remainder

E_{n}(f)

in (3.5.2) can be expanded in the form … ►About

2^{9}=512

function evaluations are needed. … ►For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). … ►For

C^{\infty}

functions Gauss quadrature can be very efficient. …

23: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: …where

A_{k}^{n}

is as in (3.3.10). … ►

B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

… ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain …

24: 27.2 Functions

… ►

27.2.9

d\left(n\right)=\sum_{d\mathbin{|}n}1

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.3 I.A
Permalink:: http://dlmf.nist.gov/27.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
…

►

25: Bibliography

… ►

W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

ⓘ

External Links:: FullText, MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

… ►

D. E. Amos (1983c) Uniform asymptotic expansions for exponential integrals

E_{n}(x)

and Bickley functions

{\rm{K}i}_{n}(x)

. ACM Trans. Math. Software 9 (4), pp. 467–479.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

ⓘ

External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

… ►

H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

\beta

- and

\gamma

-Spectroscopy:

3j

6j

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

ⓘ

External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

… ►

F. M. Arscott (1959) A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

ⓘ

External Links:: ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

…

26: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Given

P

, calculate

2P

4P

8P

by doubling as above. …If any of

2P

4P

8P

is not an integer, then the point has infinite order. Otherwise observe any equalities between

P

2P

4P

8P

, and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from

2P=-P

4P=-P

4P=-2P

8P=P

8P=-P

8P=-2P

, or

8P=-4P

. …

27: Software Index

… ► ►►►►►►►

	Open Source									With Book		Commercial
…
8 Incomplete Gamma and Related Functions
…
9 Airy and Related Functions
…
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	✓	✓	✓	✓	✓	✓		✓	✓		✓		✓	✓	a	✓
19.39(iv) $R_{C}\left(x,y\right)$ , $R_{F}\left(x,y,z\right)$ , $R_{D}\left(x,y,z\right)$ , $R_{J}\left(x,y,z,p\right)$	✓		✓			✓	✓	a	✓		✓	✓				✓	Derive
…
34 3j, 6j, 9j Symbols
…

…

28: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►

Table 18.6.1: Classical OP’s: symmetry and special values.

► ►►►

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
…
$C^{(\lambda)}_{n}\left(x\right)$	$(-1)^{n}C^{(\lambda)}_{n}\left(x\right)$	$\ifrac{{\left(2\lambda\right)_{n}}}{n!}$	$\ifrac{(-1)^{n}{\left(\lambda\right)_{n}}}{n!}$	$\ifrac{2(-1)^{n}{\left(\lambda\right)_{n+1}}}{n!}$
…

► … ►

18.6.4

\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(iv), §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

…

29: 18.9 Recurrence Relations and Derivatives

… ►

18.9.1

p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x),

ⓘ

Defines:: $A_{n}$ : coefficient (locally), $B_{n}$ : coefficient (locally) and $C_{n}$ : coefficient (locally)
Symbols:: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii), Table 18.9.1, §18.9(i)
Permalink:: http://dlmf.nist.gov/18.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

►with initial values

p_{0}(x)=1

and

p_{1}(x)=A_{0}x+B_{0}

. … ►

A_{0}

and

B_{0}

have to be understood for

\alpha+\beta=0

-1

by continuity in

\alpha

and

\beta

, that is,

A_{0}=\tfrac{1}{2}(\alpha+\beta)+1

and

B_{0}=\tfrac{1}{2}(\alpha-\beta)

. … ►

18.9.7

(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda\left(C^{(\lambda+1)}_{n}% \left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)\right),

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

►

18.9.8

4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)=-(n+1)(n+2)C^{% (\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2\lambda+1)C^{(\lambda)}_{n}% \left(x\right).

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

…

30: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8	9	10
…
8	1	8	28	56	70	56	28	8	1
9	1	9	36	84	126	126	84	36	9	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►►

$m$	$n$
$m$	…
1	1	2	3	4	5	6	7	8	9
…
8	1	9	45	165	495	1287	3003	6435	12870

► …