Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

We find two combinatorial identities on the theta series of the root lattices of the finitedimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of the well-known Essenstein series E4(z) and Ramanujan series ∆24(z). Using these two identities, we determine the theta series of certain infinite families of postive definite even unimodular lattices obtained by gluing finite copies of the root lattices of the finite-dimensional simple Lie algebras of type D2n. It turns out that these theta series are weighted symmetric polynomials of two fixed families of polynomials of E4(z) and ∆24(z).