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1: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). … ►Type II probabilistic algorithms for factoring

n

rely on finding a pseudo-random pair of integers

(x,y)

that satisfy

x^{2}\equiv y^{2}\pmod{n}

. …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …

2: 10.64 Integral Representations

… ►

Schläfli-Type Integrals

…

3: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …

4: 18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: … ► ►For type 2, with notation … ►First consider type 2. …

5: 37.21 Physical Applications

… ►When, for an optical system, the aberration function of a circular wavefront is expanded in terms of these polynomials, terms with large coefficients give insight in the type of aberration, see Born and Wolf (1999, Ch. IX). …Some low degree Zernike polynomials correspond with special types of aberrations, see Lakshminarayanan and Fleck (2011, Fig. 4). …

6: 18.36 Miscellaneous Polynomials

… ►

§18.36(i) Jacobi-Type Polynomials

… ►See Liaw et al. (2016, Eqns. 1.1 and 1.2), for the origin of this type characterization. … ►Two representative examples, type I

X_{1}

-Laguerre, Gómez-Ullate et al. (2010), and type III

X_{2}

-Hermite, Gómez-Ullate and Milson (2014) EOP’s, are illustrated here. … ►

Type I $X_{1}$ -Laguerre EOP’s

… ►

Type III $X_{2}$ -Hermite EOP’s

…

7: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials

\{p_{n}(x)\}

n=0,1,\dots

, are orthogonal on

(-1,1)

with respect to three types of weight function:

(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}

(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}

(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}

. …

8: 19.38 Approximations

… ►Approximations of the same type for

K\left(k\right)

and

E\left(k\right)

for

0<k\leq 1

are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

9: 20.8 Watson’s Expansions

… ►This reference and Bellman (1961, pp. 46–47) include other expansions of this type.

10: 16.6 Transformations of Variable

… ►

16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

ⓘ

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, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of

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

functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).