The Method Of Combinatorial Telescoping And Its Applications

The method of combinatorial telescoping is useful for proving and discovering q-series identities. The main object of this thesis is to generalize this method. We provide the method of combinatorial telescoping for sums of positive terms, while the original method can only be applied to alternating sums. Moreover, in light of the creative telescoping method for multiple sums, we also present the method of combinatorial telescoping for multiple sums.This thesis is organized as follows. The first chapter is devoted to an introduction to the background, the basic concepts and the usual notations of partition theory. We will present two of the most elemental tools for treating partitions:graphical represen-tation of partitions; infinite product generating functions. The outline of this thesis will be presented at the end of this chapter.Chapter2is devoted to the method of combinatorial telescoping presented by Chen, Hou and Sun. This method is useful for proving q-series identities, such as Watson’s identity and Sylvester’s identity. With the aid of combinatorial telescoping we shall describe a new proof of a partial theta series identity from Ramanujan’s Lost Notebook.In Chapter3, we will present the method of combinatorial telescoping for sums of positive terms, which is a variant of the method of combinatorial telescoping for alternating sums. In this chapter, we shall describe the method in detail and apply it to identities whose summands do not have the factor (-1)k. We illustrate this idea by giving a new way to prove the classical identities, such as the q-binomial theorem, MacMahon’s identity, Lebesgue’s identity as well as two identities of Yee. Besides, Andrews proposed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials. By the method of combinatorial telescoping for sums of positive terms, we establish a recurrence relation that leads to the identity of Andrews.In Chapter4, we will generalize the method of combinatorial telescoping such that it can be applied to multiple sums. We shall show the construction of the combinatorial telescoping for double sums. To illustrate this method, we shall consider two identities of Andrews. With the aid of combinatorial telescoping for double sums, we are able to prove two beautiful identities that reduce to the identities of Andrews.In Chapter5, we present a way to find a direct combinatorial interpretation via the method of combinatorial telescoping. The method of combinatorial telescoping gives an explanation of a recurrence relation, namely,0:A∪Hâ†'B∪H. By the method of cancelation presented by Feldman and Propp, the bijection φ implies a bijection ψ:Aâ†'B. More precisely, we can define the bijection ψ:Aâ†'B by setting ψ (a) to be the first element b that falls into B while iterating the action of φ on a∈A. We shall illustrate this idea by describing an involution, which leads to a proof of an identity of Andrews.Chapter6is mainly focus on parity in partition identities. Parity has played an important role in partition identities. Euler’s partition theorem states that the number of partitions of n into odd parts equals to the number of partitions of n into distinct parts. Andrews investigated a variety of parity questions in partition identities. We state an identity of Andrews, for which he ask for a combinatorial proof and we shall prove it combinatorially with the aid of Durfee square.