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 Bfunction. The $\Gamma$function is wellknown 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 objectsi.e. divided power algebras and these coefficients from this point of viewi.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?"
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