Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t

Theorem
For $n \in \Z_{\ge 0}$:


 * $\ds \sum_k \map {A_k} {r, t} \map {A_{n - k} } {s, t} = \map {A_n} {r + s, t}$

where $\map {A_n} {x, t}$ is the polynomial of degree $n$ defined as:
 * $\map {A_n} {x, t} = \dbinom {x - n t} n \dfrac x {x - n t}$

where $x \ne n t$.