Gauss%E2%80%93Christoffel%20quadrature

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11: 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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15.4.3 $$F(\frac{1}{2},1;\frac{3}{2};-z^2)=z^{-1}\arctan z,$$

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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, $\arctan z$: arctangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E3

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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12: 8.17 Incomplete Beta Functions

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8.17.7 $${\mathrm{B}}_x(a,b)=\frac{x^a}{a}F(a,1-b;a+1;x),$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.8

Permalink:

http://dlmf.nist.gov/8.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

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8.17.8 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^b}{a}F(a+b,1;a+1;x),$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

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8.17.9 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^{b-1}}{a}F(\genfrac{}{}{0pt}{}{1,1-b}{a+1};\frac{x}{x-1}).$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(ii), §8.17 and Ch.8

►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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13: 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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14: 15.16 Products

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15.16.1 $$F(\genfrac{}{}{0pt}{}{a,b}{c-\frac{1}{2}};z)F(\genfrac{}{}{0pt}{}{c-a,c-b}{c+\frac{1}{2}};z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(c\right)}_s}{{\left(c+\frac{1}{2}\right)}_s}{A}_s{z}^s,$$ ,

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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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.2 $${(1-z)}^{a+b-c}F(2a,2b;2c-1;z)=\sum _{s=0}^{\mathrm{\infty }}{A}_s{z}^s,$$ .

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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, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.3 $$F(\genfrac{}{}{0pt}{}{a,b}{c};z)F(\genfrac{}{}{0pt}{}{a,b}{c};\zeta )=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s{\left(c-a\right)}_s{\left(c-b\right)}_s}{{\left(c\right)}_s{\left(c\right)}_{2s}s!}{\left(z\zeta \right)}^{s}F(\genfrac{}{}{0pt}{}{a+s,b+s}{c+2s};z+\zeta -z\zeta ),$$ , , .

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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, ${\left(a\right)}_n$: 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 and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.4 $$F(\genfrac{}{}{0pt}{}{a,b}{c};z)F(\genfrac{}{}{0pt}{}{-a,-b}{-c};z)+\frac{ab(a-c)(b-c)}{c^2(1-c^2)}{z}^{2}F(\genfrac{}{}{0pt}{}{1+a,1+b}{2+c};z)F(\genfrac{}{}{0pt}{}{1-a,1-b}{2-c};z)=1.$$

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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, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.16

Permalink:

http://dlmf.nist.gov/15.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.5 $$F(\genfrac{}{}{0pt}{}{\frac{1}{2}+\lambda ,-\frac{1}{2}-\nu }{1+\lambda +\mu };z)F(\genfrac{}{}{0pt}{}{\frac{1}{2}-\lambda ,\frac{1}{2}+\nu }{1+\nu +\mu };1-z)+F(\genfrac{}{}{0pt}{}{\frac{1}{2}+\lambda ,\frac{1}{2}-\nu }{1+\lambda +\mu };z)F(\genfrac{}{}{0pt}{}{-\frac{1}{2}-\lambda ,\frac{1}{2}+\nu }{1+\nu +\mu };1-z)-F(\genfrac{}{}{0pt}{}{\frac{1}{2}+\lambda ,\frac{1}{2}-\nu }{1+\lambda +\mu };z)F(\genfrac{}{}{0pt}{}{\frac{1}{2}-\lambda ,\frac{1}{2}+\nu }{1+\nu +\mu };1-z)=\frac{\mathrm{\Gamma }\left(1+\lambda +\mu \right)\mathrm{\Gamma }\left(1+\nu +\mu \right)}{\mathrm{\Gamma }\left(\lambda +\mu +\nu +\frac{3}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}+\nu \right)},$$ , .

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16, §15.16 and Ch.15

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15: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:

ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

Encodings:

BibTeX

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Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.

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Notes:

Loose microfiche suppl. A1–A10

External Links:

FullText, MathReview (F. Stenger), ZentralBlatt

Referenced by:

§18.39(iii), §3.5(vi).

Encodings:

BibTeX

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16: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … ►On the circle of convergence, $|z|=1$, the Gauss series: … ►The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. … ►Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. …

17: 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).

18: 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).

19: 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).

20: 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 $k$). …