Whipple formula

SearchAdvancedHelp

(0.002 seconds)

7 matching pages

1: 14.19 Toroidal (or Ring) Functions

… ►

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

…

2: 14.9 Connection Formulas

… ►

§14.9(iv) Whipple’s Formula

…

3: Bibliography M

… ►

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

ⓘ

External Links:

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

Referenced by:

§32.8(v).

Encodings:

BibTeX

►

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

ⓘ

External Links:

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

Referenced by:

§32.9(iii).

Encodings:

BibTeX

… ►

A $q$-analog of a Whipple’s transformation for hypergeometric series in $U(n)$ . Adv. Math. 108 (1), pp. 1–76.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

►

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

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§5.18(ii).

Encodings:

BibTeX

…

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

… ►

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

… ►

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

►

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

ⓘ

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

…

5: Bibliography W

… ►

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

ⓘ

External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

Encodings:

BibTeX

… ►

Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

ⓘ

External Links:

ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§16.3(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

BibTeX

…

6: 16.4 Argument Unity

… ►

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

… ►

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)”.

… ►

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)”.

… ►

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

… ►

Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

… ►

This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

…