Pochhammer symbol

Let $n$ be a positive integer. If we denote by (x)_n:=x(x+1)...(x+n-1)

It is known that (c-b)_n/(c)_n is expressed by the hypergeometric function 2F_1.

Now let I be a subset of {0,1,2,..,n-1} and set $(x){n,I}:=\prod{i\in I}(x+i)$.

Is there a way to express (c-b){n,I}/(c){n,I} by the hypergeometric function 2F_1 or a modified version of it ?