2 added 121 characters in body

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))][k(k-a)(k-2a)...(a)]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.

1

# Variant of binomial coefficients

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))][k(k-a)(k-2a)...(a)]-1.

Has anyone this before? Or anything similar?