Binomial sum for an arbitrary function

Question

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary function and $a$ and $b$ are real and positive constants. For example for the function $f(x)=e^{x}$ the sum can be evaluated with Binomial Theorem and we have: $$S(x,a,b,n)=(e^{ax}-e^{bx})^{n}$$ But for arbitrary function $f$? Thanks.