On the ratio of gamma functions at rational arguments and its relation to Stirling numbers and discrete probability distribution

One of the most important special functions in mathematics, especially in number theory and mathematical analysis, having various applications, e.g. in such areas as probability theory, mathematical statistics, and physics, is Euler’s gamma function. The aim of this paper is two-fold. First, we consider the quotient of gamma functions at rational arguments and present the formula for it which has a simpler and more elegant expression than the combinatorial formula involving gamma function and Pochhammer symbol. Second, we obtain a limiting infinite product representation of the gamma function at rational arguments and then establish its relation to Stirling numbers of the first kind and discrete probability distribution. Some applications of the formulas are presented.