Basic Hypergeometric Series And Its Applications

In the thesis, the basic knowledge of the basic hypergeometric series, some results of the Rogers-Ramanujan type identities and its applications in number theory are studied. The main contents of this thesis can be summarized as follows:In Chapter 1, we introduce the development of the basic hypergeometric series.In Chapter 2, we introduce the basic knowledge of the basic hypergeometric series, including basic concepts, some summation formulas, transformation formulas, etc.In Chapter 3, by means of Carlitz inversion formulas and a transformation of basic hypergeometric series obtained by Chu, we give new proofs for several Rogers-Ramanujan type identities. Furthermore, some new Rogers-ramanujan type identities are obtained by a transformation of basic hypergeometric series.In Chapter 4, by some important summation formulas, we prove some results of number theory, such as Jacobi two square numbers theorem and Lagrange four square numbers theorem. In addition, by utilizing Jacobi two square numbers theorem and Lagrange four square numbers theorem and some theta function identities, we also prove the known results of number theory: two triangular number theorem, four triangular number theorem, and the number of representations of a positive integer by various quadratic forms in terms of divisor functions...