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.



Variant of binomial coefficients
(For simplicity, you probably want the last term in the numerator to be z-k+a, right? That way F(1,k)(z)=\binom{z}{k}. I'll pretend that's what you meant.)


I haven't come across such polynomials, but they're easily expressed in terms of [multifactorials](http://en.wikipedia.org/wiki/Factorial#Multifactorials). Namely,


F(a,k)(z)=z!(a)/[(z-k)!(a)k!(a)]


Note that this isn't an integer when a doesn't divide z. EDIT: (And, as Qiaochu points out, when a does divide z it's just regular ol' z-choose-k.)


