Question 99
Let $P$ be the set of all ordered pairs $(p, q)$ of nonnegative integers. Determine all functions $f: P \rightarrow \mathrm{IR}$ with the property
$$
f(p, q)=\left\{\begin{array}{c}
0 \quad \text{if} p q=0 \\
1+\frac{1}{2} f(p+1, q-1)+\frac{1}{2} f(p-1, q+1) \text{otherwise}
\end{array} .\right.
$$
$$
f(p, q)=\left\{\begin{array}{c}
0 \quad \text{if} p q=0 \\
1+\frac{1}{2} f(p+1, q-1)+\frac{1}{2} f(p-1, q+1) \text{otherwise}
\end{array} .\right.
$$