lauricella: Lauricella D-Hypergeometric Function

In multvardiv: Multivariate Probability Distributions, Statistical Divergence

Lauricella D-Hypergeometric Function

Description

Computes the Lauricella D-hypergeometric function.

Usage

lauricella(a, b, g, x, eps = 1e-06)

Arguments

a	numeric.
b	numeric vector.
g	numeric.
x	numeric vector. x must have the same length as b.
eps	numeric. Precision for the nested sums (default 1e-06).

Details

If n is the length of the b and x vectors, the Lauricella D-hypergeometric function is given by:

\displaystyle{F_D^{(n)}\left(a, b_1, ..., b_n, g; x_1, ..., x_n\right) = \sum_{m_1 \geq 0} ... \sum_{m_n \geq 0}{ \frac{ (a)_{m_1+...+m_n}(b_1)_{m_1} ... (b_n)_{m_n} }{ (g)_{m_1+...+m_n} } \frac{x_1^{m_1}}{m_1!} ... \frac{x_n^{m_n}}{m_n!} } }

where (x)_p is the Pochhammer symbol (see pochhammer).

If |x_i| < 1, i = 1, \dots, n, this sum converges. Otherwise there is an error.

The eps argument gives the required precision for its computation. It is the attr(, "epsilon") attribute of the returned value.

Value

A numeric value: the value of the Lauricella function, with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions. IEEE Signal Processing Letters, vol. 26 no. 7, July 2019. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/LSP.2019.2915000")}

N. Bouhlel and D. Rousseau (2023), Exact Rényi and Kullback-Leibler Divergences Between Multivariate t-Distributions. IEEE Signal Processing Letters, vol. 30, pp. 1672-1676, October 2023. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/LSP.2023.3324594")}

multvardiv documentation built on April 3, 2025, 6:08 p.m.