Euler transformation

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41: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials

W_{n}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

. …

42: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►

10.46.1

\phi\left(\rho,\beta;z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!\Gamma\left(% \rho k+\beta\right)},

\rho>-1

ⓘ

Defines:: $\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$ : generalized Bessel function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.46.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.46 and Ch.10

… ►The Laplace transform of

\phi\left(\rho,\beta;z\right)

can be expressed in terms of the Mittag-Leffler function: ►

10.46.3

E_{a,b}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(ak+b\right)},

a>0

ⓘ

Defines:: $E_{\NVar{a},\NVar{b}}\left(\NVar{z}\right)$ : Mittag-Leffler function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.46.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.46 and Ch.10

…

43: 15.8 Transformations of Variable

… ►

15.8.28

\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}% \right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F% \left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}% -\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+% \tfrac{1}{2}\sqrt{z}\right),

|\operatorname{ph}z|<\pi

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.8.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

…

44: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

Kummer Transformation

… ►

35.8.6

{{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}-a_{1}\right)\Gamma_{m}\left(b_{1}-a_{2}\right)}{% \Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}\right)}\*\frac{% \Gamma_{m}\left(b_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}-a_{3}% \right)}{\Gamma_{m}\left(b_{1}-a_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{2}-a% _{3}\right)}.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

►

Thomae Transformation

… ►

Laplace Transform

… ►

Euler Integral

…

45: 31.7 Relations to Other Functions

… ►

31.7.1

{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\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, $\delta$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

… ►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. … ►

\gamma=\delta=\epsilon=\tfrac{1}{2},

… ►The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

46: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates

(\xi,\eta,\phi)

, admits solutions of the form (30.13.8), where

w_{1}

satisfies the differential equation …and

w_{2}

w_{3}

satisfy (30.13.10) and (30.13.11), respectively, with

\gamma^{2}=-\kappa^{2}c^{2}\leq 0

and separation constants

\lambda

and

\mu^{2}

. Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution

z=\pm i\xi

. … ►Then

\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)

for some

n=m,m+1,m+2,\dots

, and the solution of (30.13.10) is given by (30.13.13). … ►The eigenvalues are given by

c^{2}\kappa^{2}=-\gamma^{2}

, where

\gamma^{2}

is determined from the condition …

47: 17.12 Bailey Pairs

… ►

Bailey Transform

►

17.12.1

\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},

ⓘ

Symbols:: $n$ : nonnegative integer, $\alpha_{n}$ : part of Bailey pair and $\beta_{n}$ : part of Bailey pair
Permalink:: http://dlmf.nist.gov/17.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.12, §17.12 and Ch.17

… ►

\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.

…

48: 2.6 Distributional Methods

… ►

§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of

f(t)

is defined by … ►

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

being the Mellin transform of

f(t)

or its analytic continuation (§2.5(ii)). … ►Corresponding results for the generalized Stieltjes transform … ►where

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

is the Mellin transform of

f

or its analytic continuation. …

49: Bibliography S

… ►

J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt
Referenced by:: §1.14(iii), §1.14(vii).
Encodings:: BibTeX

… ►

J. D. Secada (1999) Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

ⓘ

External Links:: ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

ⓘ

External Links:: MathReview (J. Kuntzmann), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

ⓘ

External Links:: ISSN 0021-9991, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

ⓘ

External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

…

50: Bibliography

… ►

M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

… ►

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

ⓘ

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

… ►

W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 588; package TOMS)
Referenced by:: 2nd item.
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.

ⓘ

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

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

ⓘ

External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

…