Bailey transformation of very-well-poised 8Ï•7

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21: 17.1 Special Notation

… â–ºA slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). …

â–ºThe Mellin transform of the hypergeometric function of negative argument is given by … â–ºFourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

23: 12.16 Mathematical Applications

… â–ºPCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).

24: 16.12 Products

…

25: 21.5 Modular Transformations

§21.5 Modular Transformations

â–º

§21.5(i) Riemann Theta Functions

… â–ºEquation (21.5.4) is the modular transformation property for Riemann theta functions. … â–º

§21.5(ii) Riemann Theta Functions with Characteristics

… â–ºFor explicit results in the case

g=1

, see §20.7(viii).

26: 23.15 Definitions

… â–ºAlso

\mathcal{A}

denotes a bilinear transformation on

\tau

, given by â–º

23.15.3

\mathcal{A}\tau=\frac{a\tau+b}{c\tau+d},

â“˜

Symbols:: $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $a$ : integer, $b$ : integer, $c$ : integer and $d$ : integer
Referenced by:: §20.11(i), §23.18
Permalink:: http://dlmf.nist.gov/23.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(i), §23.15 and Ch.23

… â–ºThe set of all bilinear transformations of this form is denoted by SL

(2,\mathbb{Z})

(Serre (1973, p. 77)). … â–º

23.15.5

f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),

\Im\tau>0

â“˜

Symbols:: $\Im$ : imaginary part, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $c$ : integer, $d$ : integer, $f(\tau)$ : modular function, $c_{\mathcal{A}}$ : constant and $\ell$ : level
Permalink:: http://dlmf.nist.gov/23.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(i), §23.15 and Ch.23

…

27: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

… â–ºThe Mellin transform of

f(t)

is defined by …The inversion formula is given by … â–º … â–º

§2.5(iii) Laplace Transforms with Small Parameters

…

28: 35.2 Laplace Transform

§35.2 Laplace Transform

â–º

Definition

… â–º

Inversion Formula

… â–º

Convolution Theorem

â–ºIf

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 …

29: 15.17 Mathematical Applications

… â–ºThe logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … â–ºHarmonic 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. … â–ºQuadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … â–ºBy 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. …

30: 19.15 Advantages of Symmetry

… â–ºSymmetry 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. …