DLMF: Untitled Document

… ►The monic Chebyshev polynomial

2^{1-n}T_{n}\left(x\right)

,

n\geq 1

, enjoys the ‘minimax’ property on the interval

[-1,1]

, that is,

|2^{1-n}T_{n}\left(x\right)|

has the least maximum value among all monic polynomials of degree

n

. In consequence, expansions of functions that are infinitely differentiable on

[-1,1]

in series of Chebyshev polynomials usually converge extremely rapidly. …

… ►

W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

… ►

G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

ⓘ

External Links:: ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

►

G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

ⓘ

External Links:: ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

… ►

G. Nemes (2015c) The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.

ⓘ

External Links:: ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii).
Encodings:: BibTeX

►

G. Nemes (2016) The resurgence properties of the incomplete gamma function, I. Anal. Appl. (Singap.) 14 (5), pp. 631–677.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii), §8.11(v), §8.11(v), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

…

… ►All these systems of OP’s have orthogonality properties of the form …Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … ►They are defined by their

q

-hypergeometric representations, followed by their orthogonality properties. … ►Bounds for the extreme zeros are given in Driver and Jordaan (2013). … ►Bounds for the extreme zeros are given in Driver and Jordaan (2013). …

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

Table 27.2.2: Functions related to division.

► ►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

►

… ►

S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

ⓘ

External Links:: ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.14(v).
Encodings:: BibTeX

►

S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

ⓘ

Referenced by:: §24.7(i).
Encodings:: BibTeX

… ►

J. Raynal (1979) On the definition and properties of generalized

6

-

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

ⓘ

Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… ►

M. Robnik (1980) An extremum property of the

n

-dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview
Referenced by:: §5.19(iii).
Encodings:: BibTeX

…

… ►

§8.17(i) Definitions and Basic Properties

… ►

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

►

8.17.24

I_{x}\left(m,n\right)=(1-x)^{n}\sum_{j=m}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+j-% 1}{j}x^{j},

m, n

positive integers;

0\leq x<1

.

ⓘ

Symbols:: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $n$ : nonnegative integer and $x$ : variable
A&S Ref:: 26.5.26 (The upper limit of summation has been corrected.)
Referenced by:: §8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/8.17.E24
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\infty$ .
Suggested 2011-03-23 by Stephen Bourn
See also:: Annotations for §8.17(vii), §8.17 and Ch.8

…

… ►

I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

ⓘ

Notes:: Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977
External Links:: MathReview, ZentralBlatt
Referenced by:: §2.6(i).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

Z. Gong, L. Zejda, W. Dappen, and J. M. Aparicio (2001) Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20D
External Links:: Document, Code, ZentralBlatt
Referenced by:: 7th item.
Encodings:: BibTeX

… ►

B. Grammaticos, A. Ramani, and V. Papageorgiou (1991) Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.

ⓘ

External Links:: ISSN 0031-9007, Document, MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

… ►

P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

ⓘ

External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

…

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

extremal%20properties

11—20 of 278 matching pages

11: 18.38 Mathematical Applications

12: Bibliography N

13: 18.27 $q$ -Hahn Class

14: 27.2 Functions

15: Bibliography R

16: 8.17 Incomplete Beta Functions

§8.17(i) Definitions and Basic Properties

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

17: 8 Incomplete Gamma and Related
Functions

18: 28 Mathieu Functions and Hill’s Equation

19: Bibliography G

20: 8.26 Tables

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

extremal%20properties

11—20 of 278 matching pages

11: 18.38 Mathematical Applications

12: Bibliography N

13: 18.27 q -Hahn Class

14: 27.2 Functions

15: Bibliography R

16: 8.17 Incomplete Beta Functions

§8.17(i) Definitions and Basic Properties

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

17: 8 Incomplete Gamma and Related Functions

18: 28 Mathieu Functions and Hill’s Equation

19: Bibliography G

20: 8.26 Tables

13: 18.27 $q$ -Hahn Class

17: 8 Incomplete Gamma and Related
Functions

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…