Consider the $\mathbb Z$-module that consists of the polynomials in $\mathbb Z[x,y]$ that are homogeneous polynomials of degree $d$ in the indeterminates $x$ and $y$ (homogeneous meaning that all terms of the form $c_{ij} x^i y^j$ are such that $i+j = d$, where $d$ is the degree of the polynomial). One can see that there is a unique way of writing an homogeneous polynomial of degree $d$ in the form $ \sum_{j=0}^d c_k y^{d-k} \prod_{i=0}^{k-1} (x-iy) $ because there is precisely one term where $x$ is at the $k^{th}$ power for any $k$ in the range $[0,d]$, hence we can compute coefficients. Therefore, the polynomials $y^{d-k} \prod (x-iy)$ form a basis of the $\mathbb Z$-module.
Question.
Is it possible to write any homogenous polynomial of degree $d$ in the form $ \sum_{k=0}^d c_k \left( \prod_{i=0}^{d-k-1} (y-ipx) \right) \left( \prod_{i=0}^{k-1} (x-iy) \right) $ where $p$ is a prime number? (The larger context is a number theory context, thus the prime is the thing I need. The fact that $p$ is prime might not be needed to prove this though!) Note that the polynomials formed by the 2 products are actually homogenous polynomials of degree $d$ in $x$ and $y$, so this would be a basis of the $\mathbb Z$-module of homogenous polynomials of degree $d$.
The reason for this question is because I am looking for a characterization for a certain class of polynomials that are always 0 with respect to a prime power modulus and the existence of this decomposition would help me very much. =)
If you have any suggestions please feel free to comment.