Extending the symbolic method in enumerative combinatorics. I

We use our extension of the symbolic method in enumerative combinatorics (we extend finite sums defining coefficients in generating functions to infinite series) to generalize Pólya’s theorem. This theorem determines limits of probabilities that walks in the grid graph ${\mathbb{Z}}^d$, starting at the origin, visit the given vertex. We extend ${\mathbb{Z}}^d$ to the countable complete graph $K_{\mathbb{N}}$ with weighted edges.