Function Series, Catalan Numbers, and Random Walks on Trees.

The article focuses on the connections that links Fourier series, catalan numbers and random walks on trees. Catalan numbers have many combinatorial interpretations They satisfy the recursion relations. A tree is a connected graph with no loops such that each vertex is an endpoint of only finite edges. Two distinct vertices of the tree are called neighbors if there is edge connecting them. A path in the tree is a finite or infinite sequences of vertices. It is frequently convenient to single out a specific vertex of a tree and declare it to be a root of the tree. There is a natural distance function defined on the set of vertices of a tree. Transient random walks have the property that with probability one an infinite path will visit any particular vertex only finitely many times. There is a simple test for transience of a random walk on an arbitrary tree with a nearest-neighbor probability. The generalized Catalan numbers have numerous combinatorial interpretations. The generalized Catalan numbers can be derived by means of a Pascal-like triangle.