Whipple formula

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1: 14.19 Toroidal (or Ring) Functions

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§14.19(v) Whipple’s Formula for Toroidal Functions

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2: 14.9 Connection Formulas

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§14.9(iv) Whipple’s Formula

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3: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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External Links:

ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

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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:

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

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4: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Gasper–Rahman $q$-Analog of Whipple’s ${}_3{}^{}F_2^{}$ Sum

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Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)

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17.7.11 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{n+1},c,-c}{e,c^{2}q/e,-q};q,q)=q^{\left(\genfrac{}{}{0.0pt}{}{n+1}{2}\right)}\frac{{(e{q}^{-n},e{q}^{n+1},c^2{q}^{1-n}/e,c^2{q}^{n+2}/e;q^2)}_{\mathrm{\infty }}}{{(e,c^{2}q/e;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.1.12) for Equation (17.7.11)

Permalink:

http://dlmf.nist.gov/17.7.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.12):

The missing factor $q^{\left(\genfrac{}{}{0.0pt}{}{n+1}{2}\right)}$ was inserted on the right-hand side of this summation formula.

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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5: Bibliography W

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

Encodings:

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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:

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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:

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:

FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

Encodings:

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

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6: 16.4 Argument Unity

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Whipple’s Sum

… ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ►A different type of transformation is that of Whipple: … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

7: Errata

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Just above §17.6(i) a paragraph Analytic Continuation was inserted describing the analytic continuation of the formulas which follow.

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The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

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The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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The wording was changed to make the integration variable more apparent.

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