Let $\alpha,\beta$ be two polynomials of the form $$\alpha(X)=\sum_{i=0}^{n}\alpha_iX^i,\quad \quad \beta(X)=\sum_{j=0}^n\beta_jX^j$$ where each coefficient is $1$ with a probability of $p$ and $0$ with probability $1-p$. The coefficients of $\alpha,\beta$ are all independent. Note that $\alpha,\beta$ are of degree at most $n$.
What can I say about the distribution of the polynomial $H(X)=\alpha(X)\cdot \beta (X)$?
Note that the $k$-th coefficient of $H$ is
$$h_k=\sum_{i=0}^k\alpha_i\beta_{k-i},$$
therefore it is distributed in $[0,k+1]$, but how ?
Note also that the variables $c_i^{(k)}:=\alpha_i\beta_{k-i}$ are $1$ with probability $p^2$ and $0$ with probability $1-p^2$, and are independent for fixed $k$ so I can treat $h_k$ as the sum of independent random discrete variables, which is fairly easy.
However, for different values of $k$ there is correlation between these variables, in other words $H$'s coefficients are not independent.
It looks complicated to characterize such correlation, is there any way to measure it ?
Thank you all
EDIT: As an easy example to see the correlation, note that the probability of $h_1=2$ knowing that $h_0=0$ is $0$.