§19.31 Probability Distributions

$R_G(x,y,z)$ and $R_F(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${ℝ}^2$ or ${ℝ}^3$, of the square root or reciprocal square root of a quadratic form. More generally, let $\mathbf{A}$ ($=[a_{r,s}]$) and $\mathbf{B}$ ($=[b_{r,s}]$) be real positive-definite matrices with $n$ rows and $n$ columns, and let ${\lambda }_1,\mathrm{\dots },{\lambda }_n$ be the eigenvalues of $\mathbf{A}{\mathbf{B}}^{-1}$. If $\mathbf{x}$ is a column vector with elements $x_1,x_2,\mathrm{\dots },x_n$ and transpose ${\mathbf{x}}^{\mathrm{T}}$, then

19.31.1		$${\mathbf{x}}^{\mathrm{T}}\mathbf{A}\mathbf{x}=\sum _{r=1}^n\sum _{s=1}^n{a}_{r,s}{x}_r{x}_s,$$
ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/19.31.E1 Encodings: TeX, pMML, png See also: Annotations for §19.31 and Ch.19

and

19.31.2		$${\int }_{{ℝ}^n}{({\mathbf{x}}^{\mathrm{T}}\mathbf{A}\mathbf{x})}^{\mu }\exp \left(-{\mathbf{x}}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det\mathbf{B}}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$
$\mu >-\frac{1}{2}n$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues Referenced by: §19.31 Permalink: http://dlmf.nist.gov/19.31.E2 Encodings: TeX, pMML, png See also: Annotations for §19.31 and Ch.19

§19.16(iii) shows that for $n=3$ the incomplete cases of $R_F$ and $R_G$ occur when $\mu =-1/2$ and $\mu =1/2$, respectively, while their complete cases occur when $n=2$.

For (19.31.2) and generalizations see Carlson (1972b).