Littlewood-Richardson rule gives the expansion formula for decomposing a
product of two Schur functions as a linear sum of Schur functions, while the decomposition
formula for the multiplication of two symplectic Schur function is also given by the
combinatorial method. In this paper, we will construct the algebraic forms of the
decomposition formula for the product of two symplectic Schur functions by using the
generating functions and vertex operator realizations, and then extend these results to
generalized symplectic Schur functions.