lauricella {multvardiv} | R Documentation |
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. |
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. 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. doi:10.1109/LSP.2023.3324594