Rogers-Ramanujan Identities And Q-Ultraspherical Polynomials

In this paper,we do some study on the Rogers-Ramanujan identities.Firstly,we use Carlitz inversions,q-Saalschutz identity and Heine’s transformation formulas rediscover the well-known identities:And then,we do some study on this identities,and get some result.Secondly,based on differential operator we get new identities from a exceptional case above.Finally,we introduce the theory of q-ultraspherical Polynomials and its applications in the proof of the Rogers-Ramanujan type identities.We find a formula to prove identities from Ramanujan’s Lost Notebook,Vol.1.