Example 1 Let $p, q, n \in \mathbf{N}^{+}$, find the sum: $\sum_{k=0}^{\infty} C_{p+k}^{p} \cdot C_{q+n-k}^{q}$.


The generating function of the sequence $\left\{C_{p+k}^{p}\right\}$ is $g(x)=C_{p}^{p}+C_{p+1}^{p} x+C_{p+2}^{p} x^{2}+\cdots+C_{p+k}^{p} x^{k}+\cdots=$ $\sum_{k=0}^{\infty} x^{k}=\frac{1}{(1-x)^{n+1}}$.
Similarly, the generating function of the sequence $\left\{C_{q+n-k}^{q}\right\}$ is $h(x)=\frac{1}{(1-x)^{q+1}}$, then $g(x) \cdot h(x)=\frac{1}{(1-x)^{p+1+2}}=\sum_{n=0}^{\infty}\left(\sum_{n=0}^{n} C_{p+k}^{p} C_{q+n-k}^{q}\right) x^{n}$. On the other hand, $\frac{1}{(1-x)^{p+q+2}}=\sum_{n=0}^{\infty} C_{p+q+n+1}^{p+q+1} x^{n}$. By comparing the coefficients of these two expressions, we get $\sum_{k=0}^{\infty} C_{p+k}^{p} \cdot C_{q+n-k}^{q}=C_{p+q+n+1}^{p+q+1}$.