The Beta‐Pochhammer and its application to arbitrary threshold phase error probability of a vector in Gaussian noise

Abstract

This paper develops a calculus around a new β‐Pochhammer symbol of two variables, (a,b)m,n based on the Beta weighting . The approach is a natural rising factorial formulation that offers a new way to express hypergeometric functions of two variables. The results are implemented to solve the problem of finding the phase error probability of a vector perturbed by Gaussian noise with an arbitrary phase threshold in closed form for the first time, in terms of the incomplete confluent hypergeometric function 1ℱ1. This has been an unresolved problem in angle modulation dating back to the 1950s. Closed‐form solutions are developed around the lower and upper incomplete Humbert second Φ2 confluent hypergeometric function of two variables using the new incomplete β‐Pochhammer calculus.