Whipple transformation

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1: 16.4 Argument Unity

… ►A different type of transformation is that of Whipple: …

2: Bibliography W

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X. Wang and A. K. Rathie (2013) Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

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External Links:: ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §16.6, §16.6.
Encodings:: BibTeX

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F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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3: Bibliography M

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S. C. Milne (1994) A

q

-analog of a Whipple’s transformation for hypergeometric series in

U(n)

. Adv. Math. 108 (1), pp. 1–76.

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External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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4: 16.6 Transformations of Variable

§16.6 Transformations of Variable

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Quadratic

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Cubic

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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).

5: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

§17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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F. H. Jackson’s Transformations

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Watson’s $q$ -Analog of Whipple’s Theorem

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Sears–Carlitz Transformation

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Mixed-Base Heine-Type Transformations

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