Combinatorial proofs for Jacobi’s triple product identity and Euler's pentagonal number theorem

This paper is about Jacobi’s triple product identity and Euler’s pentagonal number theorem. To be specific, it gives the proofs of Jacobi’s triple product identity via Gaussian polynomials and the q-binomial theorem, functional equation, and number theory. From this, Euler's pentagonal number theorem is derived. After that, it proves Euler's pentagonal number theorem via bijection. In addition, the paper also makes some promotions through them. For example, Jacobi's quintuple product identity is deduced from the method of Jacobi's triple product identity. The relationship between the Euler's function and the partition of integers in combinatorics. Some applications of these two formulas are also given. In particular, this paper points out their applications in physics, biology, and other fields of science, combining different branches of science closely.