Let $\Sigma$ be an alphabet of size $s$, with which we build strings of length $n$.
The Hamming ball of radius $d$ centered at $x\in\Sigma^n$ is the set $B(x, d)$ of words in $\Sigma^n$ that differ from $x$ in at most $d$ positions.
Similarly, the Hamming circle of radius $d$ centered at $x\in\Sigma^n$ is the set $C(x, d)$ of words in $\Sigma^n$ that differ from $x$ in exactly $d$ positions (note: "Hamming ball" is a standard term, but I do not know whether the expression "Hamming circle" exists at all).
Are there closed expressions for $|B(x, d_1)\cap B(y, d_2)|$ and $|C(x, d_1)\cap C(y, d_2)|$, given any two distinct strings $x,y\in\Sigma^n$ and any two $d_1,d_2\in\mathbb{N}$?