hypergeometric functions

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21: 15.1 Special Notation

… ►We use the following notations for the hypergeometric function: ►

15.1.1

{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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

►and also ►

15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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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22: 13.28 Physical Applications

… ►

§13.28(i) Exact Solutions of the Wave Equation

… ►For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). … ►

§13.28(iii) Other Applications

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23: 17.16 Mathematical Applications

§17.16 Mathematical Applications

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24: 16.23 Mathematical Applications

§16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§16.23(ii) Random Graphs

… ►

§16.23(iv) Combinatorics and Number Theory

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25: 13.27 Mathematical Applications

§13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … …

26: 16.7 Relations to Other Functions

§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).

27: 15.15 Sums

§15.15 Sums

►

15.15.1

\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(i)
Permalink:: http://dlmf.nist.gov/15.15.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §15.15 and Ch.15

… ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).

28: 35.9 Applications

§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\leq 2

and

q\leq 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). … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

29: 16 Generalized Hypergeometric Functions & Meijer G-Function

Chapter 16 Generalized Hypergeometric Functions and Meijer $G$ -Function

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30: 13 Confluent Hypergeometric Functions

Chapter 13 Confluent Hypergeometric Functions

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