Gauss formula

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11: 19.21 Connection Formulas

… ►The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). …

12: 18.3 Definitions

… ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …

13: 15.12 Asymptotic Approximations

… ►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 $0$, $j=1,2,3$. …

14: 16.12 Products

… ►

16.12.1 $${}_0{}^{}F_1^{}(-;a;z){}_0{}^{}F_1^{}(-;b;z)={}_2{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1};4z).$$

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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:

(10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8

Permalink:

http://dlmf.nist.gov/16.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

►The following formula is often referred to as Clausen’s formula ►

16.12.2 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)\right)}^2={}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b};z).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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:

§16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)

Permalink:

http://dlmf.nist.gov/16.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

… ►

16.12.3 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};z)\right)}^2=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(2a\right)}_k{\left(2b\right)}_k{\left(c-\frac{1}{2}\right)}_k}{{\left(c\right)}_k{\left(2c-1\right)}_{k}k!}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1)z^k,$$ .

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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 $n!$), $z$: 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:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

…

15: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

►For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by … … ► … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

16: 16.4 Argument Unity

… ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. …

17: 19.22 Quadratic Transformations

… ►

§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …

18: 16.16 Transformations of Variables

… ►

§16.16(i) Reduction Formulas

►

16.16.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\beta +{\beta }^{\prime };x,y)={(1-y)}^{-\alpha }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\beta +{\beta }^{\prime }};\frac{x-y}{1-y}),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\beta }^{\prime };x,y)={(1-y)}^{-\alpha }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };\frac{x}{1-y}),$$

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.5 $$F_3(\alpha ,\gamma -\alpha ;\beta ,\gamma -\beta ;\gamma ;x,y)={(1-y)}^{\alpha +\beta -\gamma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha ,\beta }{\gamma };x+y-xy),$$

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Permalink:

http://dlmf.nist.gov/16.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.16(i), §16.16 and Ch.16

… ►See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. …

19: 16.3 Derivatives and Contiguous Functions

… ►

§16.3(i) Differentiation Formulas

►

16.3.1 $$\frac{d^n}{{dz}^n}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{{\left(𝐚\right)}_n}{{\left(𝐛\right)}_n}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1+n,\mathrm{\dots },a_p+n}{b_1+n,\mathrm{\dots },b_q+n};z),$$

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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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

… ►

16.3.3 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{\gamma -1}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{\gamma ,a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)\right)={\left(\gamma \right)}_n{z}^{\gamma +n-1}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{\gamma +n,a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z),$$

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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), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p$: 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/16.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(i), §16.3 and Ch.16

… ►Two generalized hypergeometric functions ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. … ►

16.3.6 $$z{}_0{}^{}F_1^{}(-;b+1;z)+b(b-1){}_0{}^{}F_1^{}(-;b;z)-b(b-1){}_0{}^{}F_1^{}(-;b-1;z)=0,$$

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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 and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.3(ii), §16.3 and Ch.16

…

20: 18.5 Explicit Representations

… ►

§18.5(ii) Rodrigues Formulas

… ►Related formula: …See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. … ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. …