Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.24 HYPERGEOMETRIC FUNCTIONS

Definition: Confluent Hypergeometric Function The confluent hypergeometric function of the first kind is the following infinite series:

(B.130)

where {a, b} are parameters and {a(n), b(n)} are rising factorials (see Section B.22). For a, b > 0, it is represented by the following integral:

(B.131)

where is the gamma function.

It is a specific case of the generalized hypergeometric function p Fq(a; b; x) (which we do not define because the general form is not used in this book). 1F1(a; b; x) is one solution of the following second-order differential equation:

(B.132)

Comparing this equation with (B.124), we see that the Laguerre polynomials are a special case as follows:

(B.133)

The Laguerre polynomial used to describe the moments of the Rice distribution is given by Ln/2(x) = 1F1(−n/2; 1; x). The other solution to (B.132) is called the confluent hypergeometric function of the second kind, which has the following notation and integral representation: ...