Editing Dirichlet distribution (section)

===Characteristic function===

The characteristic function of the Dirichlet distribution is a [[confluent hypergeometric function|confluent]] form of the [[Lauricella hypergeometric series]].  It is given by [[Peter C. B. Phillips|Phillips]] as<ref name="phillips1988">{{cite journal |first=P. C. B. |last=Phillips |year=1988 |url=https://cowles.yale.edu/sites/default/files/files/pub/d08/d0865.pdf |title=The characteristic function of the Dirichlet and multivariate F distribution |journal=Cowles Foundation Discussion Paper 865 }}</ref>

<math display=block>
CF\left(s_1,\ldots,s_{K-1}\right) = \operatorname{E}\left(e^{i\left(s_1X_1+\cdots+s_{K-1}X_{K-1} \right)} \right)= \Psi^{\left[K-1\right]} (\alpha_1,\ldots,\alpha_{K-1};\alpha_0;is_1,\ldots, is_{K-1})
</math>

where

<math display=block>
\Psi^{[m]} (a_1,\ldots,a_m;c;z_1,\ldots z_m) = \sum\frac{(a_1)_{k_1} \cdots (a_m)_{k_m} \, z_1^{k_1} \cdots z_m^{k_m}}{(c)_k\,k_1!\cdots k_m!}.
</math>

The sum is over non-negative integers <math>k_1,\ldots,k_m</math> and <math>k=k_1+\cdots+k_m</math>.  Phillips goes on to state that this form is "inconvenient for numerical calculation" and gives an alternative in terms of a [[Methods of contour integration|complex path integral]]:

<math display=block>
\Psi^{[m]} = \frac{\Gamma(c)}{2\pi i}\int_L e^t\,t^{a_1+\cdots+a_m-c}\,\prod_{j=1}^m (t-z_j)^{-a_j} \, dt</math>

where {{mvar|L}} denotes any path in the complex plane originating at <math>-\infty</math>, encircling in the positive direction all the singularities of the integrand and returning to <math>-\infty</math>.