A Diagonal Expansion for the 2 Dirichlet Probability Density Function
P. A. Lee · SIAM Journal on Applied Mathematics · 1971
Let $X_1 $ and $X_2 $ be a pair of random variables with a joint density $p( {x_1 ,x_2 } )$, and marginal densities $p_1 ( {x_1 } )$ and $p_2 ( {x_2 } )$ respectively. If $\{ {\theta _n^{( 1 )} ( {x_1 } )} \}$ and $\{ {\theta _n^{( 2 )} ( {x_2 } )} \}$ are two sets of orthonormal polynomials, $n = 0,1,2, \cdots $ , associated with the weight functions $p_1 ( {x_1 } )$ and $p_2 ( {x_2 } )$ respectively, the diagonal expansion of Barrett and Lampard for $p( {x_1 ,x_2 } )$, if such a series exists, is given by \[ p( {x_1 ,x_2 } ) = p_1 ( {x_1 } )p_2 ( {x_2 } )\sum\limits_{n = 0}^\infty {a_n } \theta _n^{( 1 )} ( {x_1 } )\theta _n^{( 2 )} ( {x_2 } ) \] the coefficients $a_n $ being independent of both $x_1 $ and $x_2 $. Many examples of the Barrett–Lampard diagonal expansion are known. Another new example is given in this paper for the 2-variate Dirichlet probability density function defined as \[ p\left( {x_1 ,x_2 } \right) = \frac{{\Gamma \left( { u _1 + u _2 + u _3 } \right)}}{{\Gamma \left( { u _1 } \right)\Gamma \left( { u _2 } \right)\Gamma \left( { u _3 } \right)}}x_1^{ u _1 - 1} x_2^{ u _2 - 1} \left( {1 - x_1 - x_2 } \right)^{ u _3 - 1} \]where $ u _i > 0$, $i = 1,2,3$; $x_j \geqq 0$, $j = 1,2$; $x_1 + x_2 \leqq 1$. The resulting bilinear summation formula in Jacobi polynomials is believed to be new.