transformations of parameters

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21: 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

. Below are three such reductions with three and two parameters. They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … ►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. …

22: Bibliography F

… ►

A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

ⓘ

External Links:: ISSN 0024-1318, MathReview (D. A. Lutz)
Referenced by:: §32.10(vi).
Encodings:: BibTeX

…

23: 3.5 Quadrature

… ►

3.5.39

g(t)=\frac{1}{2\pi\mathrm{i}}\int_{\sigma-\mathrm{i}\infty}^{\sigma+\mathrm{i}% \infty}e^{tp}G(p)\,\mathrm{d}p,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $g(t)$ : function, $G(p)$ : Laplace transform of $g(t)$ and $\sigma$ : parameter
A&S Ref:: 29.2.1 (in different form)
Referenced by:: §3.5(viii)
Permalink:: http://dlmf.nist.gov/3.5.E39
Encodings:: pMML, png, TeX
See also:: Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

…

24: 32.8 Rational Solutions

… ►

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

–

\mbox{P}_{\mbox{\scriptsize VI}}

possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. …

25: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for

F\left(\phi,k\right)

and

E\left(\phi,k\right)

, together with special cases. ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals

K\left(k\right)

E\left(k\right)

, and

D\left(k\right)

see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

26: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. In his paper Lauricella’s hypergeometric function

F_{D}

(1963), he defined the

R

-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. If some of the parameters are equal, then the

R

-function is symmetric in the corresponding variables. This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …

27: 29.10 Lamé Functions with Imaginary Periods

… ►

29.10.1

h=\nu(\nu+1)-h^{\prime},

ⓘ

Symbols:: $h$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §29.10 and Ch.29

►

29.10.2

z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §29.10 and Ch.29

►transform (29.2.1) into ►

29.10.3

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution
Permalink:: http://dlmf.nist.gov/29.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §29.10 and Ch.29

…

28: 16.5 Integral Representations and Integrals

… ►

16.5.2

{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\,\mathrm{d}t,

\Re b_{0}>\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.4

{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}}^{t}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\,\mathrm{d}t,

c>0

\Re b_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

…

29: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►With the transformations … ► … ►

Modified Expansions

… ►

30: 19.7 Connection Formulas

… ►

Reciprocal-Modulus Transformation

… ►

Imaginary-Modulus Transformation

… ►

Imaginary-Argument Transformation

… ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). ►

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

…