F. H. Jackson transformations

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►In this subsection we let

F(x)=\mathscr{F}\left(f\right)\left(x\right)

. ►If

f(t)

is absolutely integrable on

(-\infty,\infty)

, then

F(x)

is continuous,

F(x)\to 0

x\to\pm\infty

, and … ►If

f(t)

and

g(t)

are absolutely integrable on

(-\infty,\infty)

, then so is

(f*g)(t)

, and its Fourier transform is

F(x)G(x)

, where

G(x)

is the Fourier transform of

g(t)

. … ►In this subsection we let

F_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)

F_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)

G_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0mug\mskip 3.0mu\left(x\right)

, and

G_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0mug\mskip 3.0mu\left(x\right)

. …

2: 31.7 Relations to Other Functions

… ►Other reductions of

\mathit{H\!\ell}

to a

{{}_{2}F_{1}}

, with at least one free parameter, exist iff the pair

(a,p)

takes one of a finite number of values, where

q=\alpha\beta p

. …They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). ►

31.7.2

\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

… ►Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. … ►The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

3: 33.2 Definitions and Basic Properties

… ►

§33.2(ii) Regular Solution $F_{\ell}\left(\eta,\rho\right)$

►The function

F_{\ell}\left(\eta,\rho\right)

is recessive (§2.7(iii)) at

\rho=0

, and is defined by … ►

§33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$

… ►

{H^{+}_{\ell}}\left(\eta,\rho\right)

and

{H^{-}_{\ell}}\left(\eta,\rho\right)

are complex conjugates, and their real and imaginary parts are given by … ►As in the case of

F_{\ell}\left(\eta,\rho\right)

, the solutions

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

and

G_{\ell}\left(\eta,\rho\right)

are analytic functions of

\rho

when

0<\rho<\infty

. …

4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

►

Kummer Transformation

… ►

Thomae Transformation

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. A similar result for the

{{}_{0}F_{1}}

function of matrix argument is given in Faraut and Korányi (1994, p. 346). …

5: 16.4 Argument Unity

… ►The function

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

is well-poised if … ►The function

{{}_{q+1}F_{q}}

with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For generalizations involving

{{}_{r+3}F_{r+2}}

functions see Kim et al. (2013). … ►Balanced

{{}_{4}F_{3}}\left(1\right)

series have transformation formulas and three-term relations. … ►Transformations for both balanced

{{}_{4}F_{3}}\left(1\right)

and very well-poised

{{}_{7}F_{6}}\left(1\right)

are included in Bailey (1964, pp. 56–63). …

6: 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).

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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).

7: 33.8 Continued Fractions

… ►If we denote

u=\ifrac{F_{\ell}'}{F_{\ell}}

and

p+\mathrm{i}q=\ifrac{{H^{+}_{\ell}}'}{{H^{+}_{\ell}}}

, then ►

F_{\ell}=\pm(q^{-1}(u-p)^{2}+q)^{-1/2},

►

F_{\ell}'=uF_{\ell},

►

G_{\ell}=q^{-1}(u-p)F_{\ell},

►

G_{\ell}'=q^{-1}(up-p^{2}-q^{2})F_{\ell}.

…

8: Bibliography

… ►

R. M. Aarts and A. J. E. M. Janssen (2016) Efficient approximation of the Struve functions

{\bf H}_{n}

occurring in the calculation of sound radiation quantities. The Journal of the Acoustical Society of America 140 (6), pp. 4154–4160.

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External Links:: Document
Referenced by:: §11.13(i), §11.13(i).
Encodings:: BibTeX

… ►

N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

… ►

G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:: ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

K. Aomoto (1987) Special value of the hypergeometric function

{}_{3}F_{2}

and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

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External Links:: ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

… ►

A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions

\mathbf{H}_{\nu}(x)

and

\mathbf{L}_{\nu}(x)

. J. Math. Anal. Appl. 137 (1), pp. 17–36.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt
Referenced by:: §11.4(vi), §11.7(iv).
Encodings:: BibTeX

…

9: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Integrals of the form

\int x^{\alpha}(x+t)^{\beta}F\left(a,b;c;x\right)\,\mathrm{d}x

and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). … ►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). …

10: 16.16 Transformations of Variables

§16.16 Transformations of Variables

►

§16.16(i) Reduction Formulas

… ►

§16.16(ii) Other Transformations

… ►For quadratic transformations of Appell functions see Carlson (1976).