Question

In Andrews, Askey, and Roy's Special Functions, the authors state that Gauß sums are finite field analogs of the $\Gamma$-function as Jacobi sums are to B-function. The $\Gamma$-function is well-known as the analytic continuation for the factorial function and if $x$ is in a divided power algebra then $$x^{[m]}\cdot x^{[n]}={(m+n)!\over (n!)(m!)}x^{[m+n]}.$$ where $x^{[n]}$ is the nth power of x under the divided power meaning of $x^n$. For characteristic 0, this combinatorial coefficient can be written as $${\Gamma(1+m+n)\over \Gamma(1+m)\Gamma(1+n)}=B(1+x,1+y)^{-1}\cdot[(1+x)+(1+y)].\quad (*)$$ What I mean to ask is: "Is there value in considering these objects--i.e. divided power algebras and these coefficients-- from this point of view--i.e. use of Gauß and Jacobi sums? Is this formula just a coincidence or a pattern that I want to see, or is there more to it?"