A reduction formula for the hypergeometric function using gamma random variables

Abstract

We show that the hypergeometric function _p F _q is the mean of ₀ F _q with its last argument multiplied by the product of independent gamma random variables. We use this to express for α>0 and β>0 in terms of ₁ F ₁, where I _p is the modified Bessel function. A second derivation gives the moments of the non-central chi-square random variable. Some related results are derived, including an analog of Gauss's duplication formula for _p F _q .

Keywords:

• gamma random variable
• hypergeometric function
• modified Bessel function