Spherically symmetric random walks. I. Representation in terms of orthogonal polynomials.

Spherically symmetric random walks in arbitrary dimension D can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically symmetric random walks. In general, there is a connection between orthogonal polynomials and semibounded one-dimensional random walks; such a random walk can be viewed as taking place on the set of integers n, n = 0, 1, 2, . . ., that index the polynomials. This connection allows one to express random-walk probabilities as weighted inner products of the polynomials. The correspondence between polynomials and random walks is exploited here to construct and analyze spherically symmetric random walks in D-dimensional space, where D is not restricted to be an integer. The weighted inner-product representation is used to calculate exact closed-form spatial and temporal moments of the probability distribution associated with the random walk. The polynomial representation of spherically symmetric random walks is also