Bailey bilateral summations

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11: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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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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First $q$-Analog of Bailey’s ${}_4{}^{}F_3^{}\left(1\right)$ Sum

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Second $q$-Analog of Bailey’s ${}_4{}^{}F_3^{}\left(1\right)$ Sum

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Bailey’s Nonterminating Extension of Jackson’s ${}_8{}^{}\varphi _7^{}$ Sum

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12: Bibliography

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A trinomial analogue of Bailey’s lemma and $N=2$ superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.

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External Links:

ISSN 0010-3616, Document, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

BibTeX

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Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:

In memory of Gian-Carlo Rota

External Links:

ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.

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External Links:

ISBN 0-7923-7119-4, MathReview, ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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13: Bibliography W

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A note on the trinomial analogue of Bailey’s lemma. J. Combin. Theory Ser. A 81 (1), pp. 114–118.

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External Links:

ISSN 0097-3165, Document, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

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14: Bibliography M

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

Encodings:

BibTeX

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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The $A_l$ and $C_l$ Bailey transform and lemma. Bull. Amer. Math. Soc. (N.S.) 26 (2), pp. 258–263.

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External Links:

ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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15: 5.24 Software

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Bailey (1993). Fortran and C++ wrapper.

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16: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►This reverses the order of summation in (17.6.2): … ►

Bailey–Daum $q$-Kummer Sum

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17: 16.6 Transformations of Variable

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18: 16.12 Products

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19: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

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20: 34.10 Zeros

… ►Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …