three-term 2ϕ1 transformation

(0.004 seconds)

21—30 of 813 matching pages

21: 2.5 Mellin Transform Methods

… ► … ►In the half-plane

\Re z>\max(0,-2\nu)

, the product

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.0muh% \mskip 3.0mu\left(z\right)

has a pole of order two at each positive integer, and … ►By Table 2.5.1,

\mathscr{M}\mskip-3.0muf_{2}\mskip 3.0mu\left(z\right)

is an analytic function in the half-plane

\Re z<b

. … ►Alternatively, if

\kappa\neq 0

in (2.5.18), then

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

can be continued analytically to an entire function. ►Since

\mathscr{M}\mskip-3.0muh_{1}\mskip 3.0mu\left(z\right)

is analytic for

\Re z>-c

by Table 2.5.1, the analytically-continued

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

allows us to extend the Mellin transform of

h

via …

22: 35.2 Laplace Transform

§35.2 Laplace Transform

►

Definition

… ►

Inversion Formula

… ►

Convolution Theorem

►If

g_{j}

is the Laplace transform of

f_{j}

j=1,2

, then

g_{1}g_{2}

is the Laplace transform of the convolution

f_{1}*f_{2}

, where …

23: 16.6 Transformations of Variable

§16.6 Transformations of Variable

►

Quadratic

… ►

Cubic

►

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).

24: 19.15 Advantages of Symmetry

… ►The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. ►Symmetry in

x, y, z

R_{F}\left(x,y,z\right)

R_{G}\left(x,y,z\right)

, and

R_{J}\left(x,y,z,p\right)

replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …

25: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …

26: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL

(2,\mathbb{R})

, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

27: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

►The conical functions

\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)

appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

28: 32.7 Bäcklund Transformations

… ►Then the transformations …

\mbox{P}_{\mbox{\scriptsize II}}

also has the special transformation … ►The transformations

\mathcal{S}_{j}

, for

j=1,2,3

, generate a group of order 24. … ►The quartic transformation … ►

29: 3.9 Acceleration of Convergence

… ►

§3.9(i) Sequence Transformations

… ►

§3.9(iv) Shanks’ Transformation

►Shanks’ transformation is a generalization of Aitken’s

\Delta^{2}

-process. …Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►In Table 3.9.1 values of the transforms

t_{n,2k}

are supplied for …

30: 29.21 Tables

… ►

•

Ince (1940a) tabulates the eigenvalues $a^{m}_{\nu}\left(k^{2}\right)$ , $b^{m+1}_{\nu}\left(k^{2}\right)$ (with $a^{2m+1}_{\nu}$ and $b^{2m+1}_{\nu}$ interchanged) for $k^{2}=0.1,0.5,0.9$ , $\nu=-\frac{1}{2},0(1)25$ , and $m=0,1,2,3$ . Precision is 4D.

►

•

Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$ , $n=1(1)30$ . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.