Dirichlet average

Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure, namely F(\mathbf{b};\mathbf{z})=\int f( \mathbf{u} \cdot \mathbf{z}) \, d \mu_b(\mathbf{u}), where \mathbf{u}\cdot\mathbf{z}=\sum_i^N u_i \cdot z_i and d \mu_b(\mathbf{u})=u_1^{b_1-1} \cdots u_N^{b_N-1} d\mathbf{u} is the Dirichlet measure with dimension N. They were introduced by the mathematician Bille C. Carlson in the '70s who noticed that the simple notion of this type of averaging generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:. They also play an important role for the solution of elliptic integrals (see Carlson symmetric form) and are connected to statistical applications in various ways, for example in Bayesian analysis.

Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure, namely F(\mathbf{b};\mathbf{z})=\int f( \mathbf{u} \cdot \mathbf{z}) \, d \mu_b(\mathbf{u}), where \mathbf{u}\cdot\mathbf{z}=\sum_i^N u_i \cdot z_i and d \mu_b(\mathbf{u})=u_1^{b_1-1} \cdots u_N^{b_N-1} d\mathbf{u} is the Dirichlet measure with dimension N. They were introduced by the mathematician Bille C. Carlson in the '70s who noticed that the simple notion of this type of averaging generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:. They also play an important role for the solution of elliptic integrals (see Carlson symmetric form) and are connected to statistical applications in various ways, for example in Bayesian analysis.

R-function

The (Carlson) R-function is the Dirichlet average of x^n , R_n(\mathbf{b}, \mathbf{z})=\int (\mathbf{u} \cdot \mathbf{z})^n \, d \mu_b(\mathbf{u}) with n . Sometimes R_n(\mathbf{b}, \mathbf{z}) is also denoted by R(-n;\mathbf{b}, \mathbf{z}) . Exact solutions: For n \geq 0, n \in \mathbb{N} it is possible to write an exact solution in the form of an iterative sum R_n(\mathbf{b},\mathbf{z})=\frac{\Gamma(n+1)\Gamma(b)}{\Gamma(b+n)} \cdot D_n \text{ with } D_n=\frac{1}{n}\sum_{k=1}^n \left(\sum_{i=1}^N b_i \cdot z_i^k\right) \cdot D_{n-k} where D_0=1 , N is the dimension of \mathbf{b} or \mathbf{z} and b=\sum b_i .

S-function

The (Carlson) S-function is the Dirichlet average of e^x , S(\mathbf{b}, \mathbf{z})=\int \exp(\mathbf{u} \cdot \mathbf{z}) \, d \mu_b(\mathbf{u}).

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