Gauss sum

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31: 3.5 Quadrature

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3.5.36 $$I(f)=\sum _{k=1}^n{w}_{k}f({\zeta }_k)+E_n(f),$$

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Symbols:

$I(f)$: Bromwich integral, ${\zeta }_k$: Gauss nodes, $E_n(f)$: error term and $w_k$: weights

Referenced by:

§3.5(viii), §3.5(viii)

Permalink:

http://dlmf.nist.gov/3.5.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5 and Ch.3

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3.5.41 $$g(t)=\sum _{k=1}^n\frac{w_k{\zeta }_k}{\sqrt{{\zeta }_k^2+t^2}},$$

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Symbols:

${\zeta }_k$: Gauss nodes, $g(t)$: function and $w_k$: weights

Permalink:

http://dlmf.nist.gov/3.5.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

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32: 16.23 Mathematical Applications

… ►In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. … ►The Bieberbach conjecture states that if ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is a conformal map of the unit disk to any complex domain, then $|a_n|\le n|a_1|$. … ►

16.23.1 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};x)>0,$$

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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

Permalink:

http://dlmf.nist.gov/16.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.23(iii), §16.23 and Ch.16

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33: 13.31 Approximations

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13.31.1 $$A_n(z)=\sum _{s=0}^n\frac{{\left(-n\right)}_s{\left(n+1\right)}_s{\left(a\right)}_s{\left(b\right)}_s}{{\left(a+1\right)}_s{\left(b+1\right)}_s{(n!)}^2}{}_3{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-n+s,n+1+s,1}{1+s,a+1+s,b+1+s};-z),$$

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Defines:

$A_n(z)$: expansion (locally)

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 $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

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34: 16.8 Differential Equations

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16.8.8 $${}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $!$: factorial (as in $n!$), $q$: 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

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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35: 18.12 Generating Functions

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18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3 $${(1+z)}^{-\alpha -\beta -1}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1),\frac{1}{2}(\alpha +\beta +2)}{\beta +1};\frac{2(x+1)z}{{(1+z)}^2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\alpha +\beta +1\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.12.3_5), §18.12, §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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36: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a_1\right]}_{\kappa }\mathrm{\cdots }{\left[a_p\right]}_{\kappa }}{{\left[b_1\right]}_{\kappa }\mathrm{\cdots }{\left[b_q\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Defines:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument

Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer, $p$: nonnegative integer, $q$: nonnegative integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.10, §35.10, §35.8(i), §35.8(i), §35.8(i), §35.8(i)

Permalink:

http://dlmf.nist.gov/35.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(i), §35.8 and Ch.35

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37: 16.16 Transformations of Variables

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16.16.7 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k{\left(\alpha +\beta -\gamma -{\gamma }^{\prime }+1\right)}_k}{{\left(\gamma \right)}_k{\left({\gamma }^{\prime }\right)}_{k}k!}{x}^k{y}^k{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{\gamma +k};x){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k}{{\gamma }^{\prime }+k};y);$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.16.E7

Encodings:

pMML, png, TeX

See also:

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

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38: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.13 $${\varphi }_n(z)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +1\right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}{z}^{\mathrm{\ell }}=\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda +1}{-\lambda -n+1};z),$$

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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), $!$: factorial (as in $n!$), $z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

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39: 16.11 Asymptotic Expansions

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16.11.10 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1+r,\mathrm{\dots },a_{k-1}+r,a_k,\mathrm{\dots },a_{p+1}}{b_1+r,\mathrm{\dots },b_k+r,b_{k+1},\mathrm{\dots },b_p};z)=\sum _{n=0}^{m-1}\frac{{\left(a_1+r\right)}_n\mathrm{\cdots }{\left(a_{k-1}+r\right)}_n{\left(a_k\right)}_n\mathrm{\cdots }{\left(a_{p+1}\right)}_n}{{\left(b_1+r\right)}_n\mathrm{\cdots }{\left(b_k+r\right)}_n{\left(b_{k+1}\right)}_n\mathrm{\cdots }{\left(b_p\right)}_n}\frac{z^n}{n!}+O\left(\frac{1}{r^m}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $r$: large parameter

Permalink:

http://dlmf.nist.gov/16.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(iii), §16.11 and Ch.16

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16.11.11 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1+r,\mathrm{\dots },a_p+r}{b_1+r,\mathrm{\dots },b_q+r};z)=\sum _{n=0}^{m-1}\frac{{\left(a_1+r\right)}_n\mathrm{\cdots }{\left(a_p+r\right)}_n}{{\left(b_1+r\right)}_n\mathrm{\cdots }{\left(b_q+r\right)}_n}\frac{z^n}{n!}+O\left(\frac{1}{r^{(q-p)m}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $r$: large parameter

Permalink:

http://dlmf.nist.gov/16.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(iii), §16.11 and Ch.16

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40: 15.4 Special Cases

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15.4.1 $$F(1,1;2;z)=-z^{-1}\ln \left(1-z\right),$$

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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, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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15.4.2 $$F(\frac{1}{2},1;\frac{3}{2};z^2)=\frac{1}{2z}\ln \left(\frac{1+z}{1-z}\right),$$

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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, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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$$F(a,b;a;z)={(1-z)}^{-b},$$

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$$F(a,b;b;z)={(1-z)}^{-a},$$

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Dougall’s Bilateral Sum

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