- #1
winterfors
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Given a http://en.wikipedia.org/wiki/Simplex#The_standard_simplex" in \mathbb{R}^n:
\Omega = \left\{ {{\textbf{y}}:\,\,\,\,{y_i} \ge 0,\,\,\,\,1 = \sum\limits_{i = 1}^n {{y_i}} } \right\}
where n is a positive integer, and a vector \textbf{a} with n elements a_i>1,
I need to evaluate the integral
(1) 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}
where t and s are real non-negative numbers, and m is an integer between 1 and n. The integral is taken over the standard n-1 simplex \Omega with respect to dy = dy_1 dy_2...dy_n.
Similar integrals that have well-known solutions are:
(2) \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)}}
and
(3) \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)}}
where \Gamma is the Gamma function.
Note that the only difference between (1) and (3) is that the second sum in (3) goes from m+1 to n instead of from m to n. This difference is however important, surely making the solution to (1) more complicated than that of (3).
\Omega = \left\{ {{\textbf{y}}:\,\,\,\,{y_i} \ge 0,\,\,\,\,1 = \sum\limits_{i = 1}^n {{y_i}} } \right\}
where n is a positive integer, and a vector \textbf{a} with n elements a_i>1,
I need to evaluate the integral
(1) 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}
where t and s are real non-negative numbers, and m is an integer between 1 and n. The integral is taken over the standard n-1 simplex \Omega with respect to dy = dy_1 dy_2...dy_n.
Similar integrals that have well-known solutions are:
(2) \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)}}
and
(3) \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)}}
where \Gamma is the Gamma function.
Note that the only difference between (1) and (3) is that the second sum in (3) goes from m+1 to n instead of from m to n. This difference is however important, surely making the solution to (1) more complicated than that of (3).
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