How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
$$f:(\mathbb{N}\cup\{0\})\times(\mathbb{N}\cup\{0\})\rightarrow[0,1]$$ Base case: $f(0,b)=(1-np)^b$ $\forall$ $b\geq 0$.
$f(a,b)=f(a-1,b-1)(n-a+1)p+f(a,b-1)[1-(n-a)p]$, f(a,b)=f(a-1,b-1)[n-(a-1)]p+f(a,b-1)[1-(n-a)p]$, if $a\leq \min{(n,b)}$.
$f(a,b)=0$, otherwise.
Note: The recursive definition could just be applied for $a\leq b$. The function values for $n< a\leq b$ (if $n < b$) can be set to $0$ laterafterwards.
I have attempted using a multivariate generating function, but it doesn't yield a good simplification. Also looked into somehow manipulating the formula for multivariate taylor series, but to no avail.
