Gauss formula

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21: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. … ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). These references all use results related to the integral formulas (35.4.7) and (35.5.8). …

22: 15.5 Derivatives and Contiguous Functions

… ►

§15.5(i) Differentiation Formulas

… ►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►

15.5.12 $$(b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z)=0,$$

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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

Permalink:

http://dlmf.nist.gov/15.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

… ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. … ►

15.5.20 $$z(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),$$

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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, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

…

23: 15.6 Integral Representations

… ►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: ►

15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.3 $$𝐅(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_{\mathrm{\infty }}^{(0+)}\frac{t^{b-1}{(t+1)}^{a-c}}{{(t-zt+1)}^a}dt,$$ ; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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

Keywords:

Mellin transform

Referenced by:

§15.6, §15.6

Permalink:

http://dlmf.nist.gov/15.6.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.8 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1𝐅(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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

Keywords:

Mellin transform

Proof sketch:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

…

24: 18.35 Pollaczek Polynomials

… ►

18.35.4 $$P_n^{(\lambda )}(\cos \theta ;a,b)=\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_n}{n!}{\mathrm{e}}^{\mathrm{i}n\theta }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{-n-\lambda +1+\mathrm{i}{\tau }_{a,b}(\theta )};{\mathrm{e}}^{-2\mathrm{i}\theta })=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta },$$

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Defines:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial

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), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.10)); In the cited formula, in the lower parameter of the hypergeometric function, $-\lambda$ should be replaced by $-\lambda +1$.(proved)

Referenced by:

(18.35.4_5)

Permalink:

http://dlmf.nist.gov/18.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(i), §18.35 and Ch.18

…

25: Bibliography V

… ►

Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

… ►

Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:

ISSN 0019-5839, MathReview Entry, ZentralBlatt

Referenced by:

(17.6.4_5), §17.6(i), (17.8.8), §17.8.

Encodings:

BibTeX

… ►

Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

…

26: 15.10 Hypergeometric Differential Equation

… ►

$f_1(z)=F({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z),$

… ►

$f_1(z)=F({\displaystyle \genfrac{}{}{0pt}{}{a,b}{a+b+1-c}};1-z),$

… ►(b) If $c$ equals $n=1,2,3,\mathrm{\dots }$, and $a\ne 1,2,\mathrm{\dots },n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by $F(a,b;n;z)$ and … ►

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

27: 15.8 Transformations of Variable

… ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). … ►

15.8.13 $$F(\genfrac{}{}{0pt}{}{a,b}{2b};z)={\left(1-\frac{1}{2}z\right)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{b+\frac{1}{2}};{\left(\frac{z}{2-z}\right)}^2),$$ ,

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

Referenced by:

§15.8(iv)

Permalink:

http://dlmf.nist.gov/15.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.14 $$F(\genfrac{}{}{0pt}{}{a,b}{2b};z)={\left(1-z\right)}^{-a/2}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,b-\frac{1}{2}a}{b+\frac{1}{2}};\frac{z^2}{4z-4}),$$ .

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

Permalink:

http://dlmf.nist.gov/15.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►

15.8.15 $$F(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)={(1+z)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{a-b+1};\frac{4z}{{(1+z)}^2}),$$ ,

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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 and $b$: real or complex parameter

Referenced by:

§15.4(i), §15.8(iv)

Permalink:

http://dlmf.nist.gov/15.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.16 $$F(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)={(1-z)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}}{a-b+1};\frac{-4z}{{(1-z)}^2}),$$ .

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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 and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

…

28: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

… ►

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

… ►

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

… ►

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

… ►

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

… ►

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

…

29: Bibliography M

… ►

A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

BibTeX

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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External Links:

ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

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

… ►

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

… ►

The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

…

30: 13.29 Methods of Computation

… ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►Gauss quadrature methods are discussed in Gautschi (2002b). …