Hello i have given recurrence like this :
$$p_{n,k}=(1-q)p_{n-1,k-1}+qp_{n-1,k}$$
my question is how to get (step by step) generating function from this recurrence?
we know that it's some king of distribution and from how it looks we can say it's binomial distribution.
The probability generating function is $G_n(t):=\sum_{k=0}^n p_{n,\,k}t^k$, so $G_0=1$ and for $n\ge 1$ we have $$G_n(t)=q^n+\sum_{k=1}^n ((1-q)p_{n-1,\,k-1}+qp_{n-1,\,k})t^k\\=q^n+(1-q)tG_{n-1}(t)+q(G_{n-1}(t)-q^{n-1})\\=(q+(1-q)t)G_{n-1}(t).$$Hence $G_n(t)=(q+(1-q)t)^n$ for all $n$.