I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc.

First recall the following. If z is a formal variable, then we can consider \binom{z}{k} as a polynomial in z by the standard formula: \binom{z}{k}= [z(z-1)...(z-k+1))]\*[k!]-1.

Here's the variant I came across. Let a and k integers, where a divides k, and we write k=ab. Consider the polynomial 

F(a,k)(z)=[z(z-a)(z-2a)...(z-k+a))][k(k-a)(k-2a)...(a)]-1.

Has anyone this before? Or anything similar?

**Update:** Jonah helpfully identifies a typo in the numerator of F(a,k)(z) which I've now corrected.