Incomplete Exponential and Hypergeometric Functions with Applications to the Non Central χ ² -Distribution

Corresponding to the incomplete gamma functions,found useful in many problems,we propose incomplete exponential functions. Like the generalized incomplete gamma functions,the proposed functions have an additional parameter. It is shown that these functions can be related to Bessel functions. This leads us naturally to an incomplete extension of the hypergeometric function. The usefulness of these functions in some closed form representation of the non central χ2-distributions is demonstrated.