  In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which
are conjectured to have important information about the characters of
irreducible modules of a reductive group over a field of positive
characteristic, and also about those of an affine Kac-Moody algebra at the
critical level. The periodic Kazhdan-Lusztig polynomials can be computed by
using another family of polynomials, called the periodic $R$-polynomials. In
this paper, we prove a (closed) combinatorial formula expressing periodic
$R$-polynomials in terms of the "doubled" Bruhat graph associated to a finite
Weyl group and a finite root system.
