  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 $n$, $n=0,~1,~2,~\ldots$, 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 {\sl 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 used to calculate the two-point Green's function for a
rotationally symmetric free scalar quantum field theory.
