Kummer transformations

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1—10 of 24 matching pages

1: 16.6 Transformations of Variable

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

â–ºFor 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).

2: 13.10 Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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§13.10(v) Hankel Transforms

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3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Kummer Transformation

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4: 13.2 Definitions and Basic Properties

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Kummer’s Transformations

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5: Bibliography P

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R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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6: 13.9 Zeros

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§13.9(i) Zeros of $M\left(a,b,z\right)$

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§13.9(ii) Zeros of $U\left(a,b,z\right)$

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7: 16.4 Argument Unity

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§16.4(iii) Identities

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8: 33.14 Definitions and Basic Properties

… â–ºThis is a consequence of Kummer’s transformation (§13.2(vii)). …

9: 33.2 Definitions and Basic Properties

… â–ºThis is a consequence of Kummer’s transformation (§13.2(vii)). …

10: 13.8 Asymptotic Approximations for Large Parameters

… â–ºWhen the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when

|b|

is large, and

|b-a|

and

|z|

are bounded. …

Kummer transformations

1—10 of 24 matching pages

1: 16.6 Transformations of Variable

2: 13.10 Integrals

§13.10(ii) Laplace Transforms

§13.10(iii) Mellin Transforms

§13.10(iv) Fourier Transforms

§13.10(v) Hankel Transforms

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

Kummer Transformation

4: 13.2 Definitions and Basic Properties

Kummer’s Transformations

5: Bibliography P

6: 13.9 Zeros

§13.9(i) Zeros of M â¡ ( a , b , z )

§13.9(ii) Zeros of U â¡ ( a , b , z )

7: 16.4 Argument Unity

§16.4(iii) Identities

8: 33.14 Definitions and Basic Properties

9: 33.2 Definitions and Basic Properties

10: 13.8 Asymptotic Approximations for Large Parameters

§13.9(i) Zeros of $M\left(a,b,z\right)$

§13.9(ii) Zeros of $U\left(a,b,z\right)$