power series expansion in two variables

Question

Let $p,q \in (0,1)$ sucht that $p + q = 1$. For $x_{1},x_{2} \in (-1,1)$ define the function $g(x_{1},x_{2})$ by $$ g(x_{1},x_{2}) = \frac{qx_{1} + p^{2}x_{1}x_{2}}{1 - pqx_{1}x_{2}}. $$ Now $g$ is the generating function of some probability distribution $q(n_{1},n_{2})$ for $n_{1},n_{2} \geq 0$,i.e. $$ \sum_{n_{1},n_{2}=0}^{\infty}q(n_{1},n_{2})x_{1}^{n_{1}}x_{2}^{n_{2}} = g(x_{1},x_{2}). $$ How can I expand $g$, so I can determine $q$.

Answer 1

Since the constraints that you give result into $$|pq x_1 x_2|<1$$ then you can apply the geometric series to the denominator and the expansion is simply $$ \eqalign{ & \left( {q\,x_1 + p^2 x_1 x_2 } \right)\sum\limits_{0 \le k} {\left( {pq\,x_1 x_2 } \right)^{\,k} } = \cr & = \sum\limits_{0 \le k} {p^{\,k} q^{\,k + 1} \,x_1 ^{\,k + 1} x_2 ^{\,k} + p^{\,k + 2} q^{\,k} \,x_1 ^{\,k + 1} x_2 ^{\,k + 1} } = \cr & = \sum\limits_{0 \le k} {\left( {p^{\,k} q^{\,k + 1} \,x_2 ^{\,k} + p^{\,k + 2} q^{\,k} \,x_2 ^{\,k + 1} } \right)x_1 ^{\,k + 1} } = \cr & = \sum\limits_{0 \le n_1 } {\sum\limits_{0 \le n_2 } {c\left( {n_1 ,n_2 } \right)x_1 ^{\,n_{\,1} } x_2 ^{\,n_{\,2} } } } \cr & \quad \quad \quad \Downarrow \cr & c\left( {n_1 ,n_2 } \right) = \left[ {1 \le n_1 } \right]\left( {\left[ {n_2 = n_1 - 1} \right]\left( {p^{\,n_1 - 1} q^{\,n_1 } } \right) + \left[ {n_2 = n_1 } \right]p^{\,n_1 + 1} q^{\,n_1 - 1} } \right) \cr} $$

where $[P]$ denotes the Iverson bracket $$ \left[ P \right] = \left\{ {\begin{array}{*{20}c} 1 & {P = TRUE} \\ 0 & {P = FALSE} \\ \end{array} } \right. $$