On The Proofs Of Rogers-Ramanujan Identities And The Q^-1-Hermite Polynomials

The history of q-series has been more than two hundreds and the proof of the Rogers-Ramanujan identities is always one focus of this theory. Kinds of ways have been used to prove these identities.The main results of this paper are following:Starting from Cauchy q-binomial theorem and two other q-identities, we compute an important q-beta integral. By this integral and Jacobi triple product identity, a new proof of the Rogers-Ramanujan identities is given.We study q^-1-Hermite polynomials by q-differential operator and obtain some properties of these polynomials.Using the orthogonality relation of q-Hermite polynomials and the relation between q-Hermite polynomials and q^-1-Hermite polynomials, we derive some identities of the Rogers-Ramanujan type.