Given a http://en.wikipedia.org/wiki/Simplex#The_standard_simplex" in [itex]\mathbb{R}^n[/itex]:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Omega = \left\{ {{\textbf{y}}:\,\,\,\,{y_i} \ge 0,\,\,\,\,1 = \sum\limits_{i = 1}^n {{y_i}} } \right\}[/tex]

wherenis a positive integer, and a vector [itex]\textbf{a}[/itex] withnelements [itex]a_i>1[/itex],

I need to evaluate the integral

(1) [tex]G({\textbf{a}},m,s,t) = \int\limits_{\textbf{y} \in \Omega } {{{\left( {\sum\limits_{i = 1}^m {{y_i}} } \right)}^s}{{\left( {\sum\limits_{i = m}^n {{y_i}} } \right)}^t}\left( {\prod\limits_{i = 1}^n {{y_i}^{{a_i} - 1}} } \right)dy} [/tex]

wheretandsare real non-negative numbers, andmis an integer between 1 andn. The integral is taken over the standardn-1 simplex [itex]\Omega[/itex] with respect to [itex]dy = dy_1 dy_2...dy_n[/itex].

Similar integrals that have well-known solutions are:

(2) [tex]\int\limits_{y \in \Omega } {\prod\limits_{i = 1}^n {{y_i}^{{a_i} - 1}} dy} = \frac{{\prod\limits_{i = 1}^n {\Gamma ({a_i})} }}{{\Gamma \left( {\sum\limits_{i = 1}^n {{a_i}} } \right)}}[/tex]

and

(3) [tex]\int\limits_{y \in \Omega } {{{\left( {\sum\limits_{i = 1}^m {{y_i}} } \right)}^s}{{\left( {\sum\limits_{i = m + 1}^n {{y_i}} } \right)}^t}\left( {\prod\limits_{i = 1}^n {{y_i}^{{a_i} - 1}} } \right)dy} = \frac{{\prod\limits_{i = 1}^n {\Gamma ({a_i})} }}{{\Gamma \left( {s + t + \sum\limits_{i = 1}^n {{a_i}} } \right)}}\frac{{\Gamma \left( {s + \sum\limits_{i = 1}^m {{a_i}} } \right)}}{{\Gamma \left( {\sum\limits_{i = 1}^m {{a_i}} } \right)}}\frac{{\Gamma \left( {t + \sum\limits_{i = m + 1}^n {{a_i}} } \right)}}{{\Gamma \left( {\sum\limits_{i = m + 1}^n {{a_i}} } \right)}}[/tex]

where [itex]\Gamma[/itex] is the Gamma function.

Note that the only difference between (1) and (3) is that the second sum in (3) goes fromm+1 toninstead of frommton. This difference is however important, surely making the solution to (1) more complicated than that of (3).

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# Integral over standard n-simplex

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