Rogers-Ramanujan Identities And Q-Operator

The history of q-series has been more than two hundred,Rogers-Ramanujan type identities always play a central role in the theory of q-series. British mathematician Rogers and Indian mathematician Ramanujan have discovered famous Rogers-Ramanujan identities respectively. In the thesis, we start from two well-known Rogers-Ramanujan identities and q-ultraspherical Polynomials, some Rogers-Ramanujan type identities can be arrived. Using Sears-Transformation, we can get some Rogers-Ramanujan type identities.Then, we use q-Leibniz rule to prove some important transformations in different ways, including q-Binomial theorem, q-Chu-Vandermonde summation.