The Research Of Several Restricted Partition Functions

The theory of integer partitions is a fascinating branch of combinatorics,in which the restricted partition functions has attracted many scholars.In this thesis,we study three restricted partition functions and obtain their generating functions.In the first part,we study k-regular partitions with difference at most kt between largest and smallest parts,and obtain the generating function by using q-series and combinatorial methods,respectively.We also use(q,z)-over Gaussian polynomials to prove the generating functions of overpartitions with bounded part differences.In the second part,based on Andrews' work on Semi-Fibonacci partitions,we introduce the definition of semi-(m,j)-Fibonacci partitions,and prove that it's partition function is equal to the partition function of m-power partition with restrictions.In the third part,we focus on the number of parts which is divisible by k of the partitions of n wherein each part repeated less than k times.And we also prove a identity by two methods.