Gauss%E2%80%93Legendre formula

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31: Bibliography

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-62321-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§1.10(vii), §1.15(iii), §1.15(vi), §1.15(vii), §10.22(vi), §13.6(v), §13.9(i), Ch.13, §14.3(iii), §14.30(iv), §15.10(i), §15.11(i), §15.11(ii), §15.14, §15.17(iv), §15.2(i), §15.4(ii), §15.5(i), §15.5(ii), §15.6, §15.7, §15.8(iv), §15.8(iii), Ch.15, §16.4(ii), §16.4(iii), §16.4(v), §16.4(v), Ch.16, §17.12, Ch.17, §18.10(ii), §18.10(iv), §18.11(i), §18.12, (18.17.41_5), §18.17(iv), §18.17(viii), §18.18(iv), §18.18(iv), §18.18(v), §18.18(vii), §18.2(i), §18.2(iv), §18.2(vi), §18.20(ii), §18.22(i), §18.25(ii), §18.26(i), Table 18.3.1, §18.38(ii), §18.5(iii), (18.7.25), Ch.18, §26.19, §4.10(i), §4.10(ii), §5.13, §5.14, §5.16, §5.16, §5.18(i), §5.18(ii), §5.18(iii), §5.20, §5.4(ii), §5.5(iv), Ch.5, Ch.6, Profile Richard A. Askey, Profile Ranjan Roy, Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

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32: 15.1 Special Notation

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15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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33: 16.6 Transformations of Variable

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16.6.1 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{a-b+1,a-c+1};z)={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)}{a-b+1,a-c+1};\frac{-4z}{{(1-z)}^2}).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

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16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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).

34: 15.12 Asymptotic Approximations

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15.12.2 $$F(a,b;c;z)=\sum _{s=0}^{m-1}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s+O\left(c^{-m}\right),$$ $|c|\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

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15.12.3 $$F(\genfrac{}{}{0pt}{}{a,b}{c+\lambda };z)\sim \frac{\mathrm{\Gamma }\left(c+\lambda \right)}{\mathrm{\Gamma }\left(c-b+\lambda \right)}\sum _{s=0}^{\mathrm{\infty }}{q}_s(z){\left(b\right)}_s{\lambda }^{-s-b},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $q_s(z)$: coefficients

Referenced by:

§15.12(iii), §15.12(iii), §15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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15.12.7 $$F(\genfrac{}{}{0pt}{}{a,b-\lambda }{c+\lambda };-z)=2^{b-c+(1/2)}{\left(\frac{z+1}{2\sqrt{z}}\right)}^{\lambda }\left({\lambda }^{a/2}U(a-\frac{1}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}+O({\lambda }^{-1})\right)+\frac{{\lambda }^{(a-1)/2}}{\alpha }U(a-\frac{3}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}-2^{c-b-(1/2)}{\left(\frac{\alpha }{z-1}\right)}^a+O({\lambda }^{-1})\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\alpha$

Permalink:

http://dlmf.nist.gov/15.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. …

35: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

36: 31.11 Expansions in Series of Hypergeometric Functions

… ►The formulas in this section are given in Svartholm (1939) and Erdélyi (1942b, 1944). … ►

31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

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Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

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37: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

§16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

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16.10.1 $${}_{p+r}{}^{}F_{q+s}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,c_1,\mathrm{\dots },c_r}{b_1,\mathrm{\dots },b_q,d_1,\mathrm{\dots },d_s};z\zeta )=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(𝐚\right)}_k{\left(\alpha \right)}_k{\left(\beta \right)}_k{(-z)}^k}{{\left(𝐛\right)}_k{\left(\gamma +k\right)}_{k}k!}{}_{p+2}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k,a_1+k,\mathrm{\dots },a_p+k}{\gamma +2k+1,b_1+k,\mathrm{\dots },b_q+k};z){}_{r+2}{}^{}F_{s+2}^{}(\genfrac{}{}{0pt}{}{-k,\gamma +k,c_1,\mathrm{\dots },c_r}{\alpha ,\beta ,d_1,\mathrm{\dots },d_s};\zeta ).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

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16.10.2 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};z\zeta )={(1-z)}^{-a_1}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k}{k!}{}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-k,a_2,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};\zeta ){\left(\frac{z}{z-1}\right)}^k.$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

… ►Expansions of the form ${\sum }_{n=1}^{\mathrm{\infty }}{(\pm 1)}^n{}_p{}^{}F_{p+1}^{}(𝐚;𝐛;-n^2{z}^2)$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

38: 27.10 Periodic Number-Theoretic Functions

… ►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. …In particular, $G(n,{\chi }_1)=c_k\left(n\right)$. ► $G(n,\chi )$ is separable for some $n$ if … ►For a primitive character $\chi (modk)$, $G(n,\chi )$ is separable for every $n$, and … ►Conversely, if $G(n,\chi )$ is separable for every $n$, then $\chi$ is primitive (mod $k$). …

39: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►Equivalently, the function is denoted by ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ or ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$, and sometimes, for brevity, by ${}_p{}^{}F_q^{}\left(z\right)$. … ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). … ►

16.2.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{-m,𝐚}{𝐛};z)=\frac{{\left(𝐚\right)}_m{(-z)}^m}{{\left(𝐛\right)}_m}{}_{q+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-m,1-m-𝐛}{1-m-𝐚};\frac{{(-1)}^{p+q}}{z})$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $p$: nonnegative integer, $q$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/16.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►See §16.5 for the definition of ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. …

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

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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

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

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