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

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 1 ℱ 1 . 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 Φ 2 confluent hypergeometric function of two variables using the new incomplete β‐Pochhammer calculus.