Dirichlet average


Dirichlet averages are averages of functions under the Dirichlet density. An important one are dirichlet averages that have a certain argument structure, namely
where and 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 and are connected to statistical applications in various ways, for example in Bayesian analysis.

Notable Dirichlet averages

Some Dirichlet averages are so fundamental that they are named. A few are listed below.

R-function

The R-function is the Dirichlet average of,
with. Sometimes is also denoted by.
Exact solutions:
For it is possible to write an exact solution in the form of an iterative sum
where, is the dimension of or and.

S-function

The S-function is the Dirichlet average of,