  We introduce a family of rings of symmetric functions depending on an
infinite sequence of parameters. A distinguished basis of such a ring is
comprised by analogues of the Schur functions. The corresponding structure
coefficients are polynomials in the parameters which we call the
Littlewood-Richardson polynomials. We give a combinatorial rule for their
calculation by modifying an earlier result of B. Sagan and the author. The new
rule provides a formula for these polynomials which is manifestly positive in
the sense of W. Graham. We apply this formula for the calculation of the
product of equivariant Schubert classes on Grassmannians which implies a
stability property of the structure coefficients. The first manifestly positive
formula for such an expansion was given by A. Knutson and T. Tao by using
combinatorics of puzzles while the stability property was not apparent from
that formula. We also use the Littlewood-Richardson polynomials to describe the
multiplication rule in the algebra of the Casimir elements for the general
linear Lie algebra in the basis of the quantum immanants constructed by A.
Okounkov and G. Olshanski.
