Generalized Gosper’s Algorithm And Its Applications

We mainly consider the extension of Gosper’s algorithm in dealing with summations of unspecific sequences.Our first goal is to extend the research object of Gosper’s algorithm,present a generalized Gosper’s algorithm to solve the telescoping of general sequences.Along this approach,we provide a new proof of Ma’s summation formula and propose a new summation formula.Our second goal is to use the generalized Gosper’s algorithm to provide a new inversion formula which implies several classic inversion pairs.Finally,inspired by Karr’s algorithm,we consider the summations involving a sequence satisfying a recurrence of order two.We transform the summation problem into solving the difference equation,and present the algorithm of solving the difference equation.The thesis is organized as follows.The first chapter briefly introduces the indefinite sum problem,especially the research background and literature on the indefinite sum problem inspired by Gosper’s algorithm.Chapter 2 reviews the key ideas and full algorithm of Gosper’s algorithm.Chapter 3 extends Gosper’s algorithm to sequences beyond the hypergeometric ones.Based on Gosper’s algorithm,this chapter presents an approach to the telescoping of general sequences.Along this approach,it gives a new proof of Ma’s summation formula,and proposes a new summation formula for the generalized function.Also this chapter shows a bibasic extension of Ma’s inversion formula by using the new summation formula.From the formulas,it is able to derive several hypergeometric and elliptic hypergeometric identities,and several inversion formulas.Chapter 4 considers the summations involving a sequence satisfying a recurrence of order two.Inspired Karr’s algorithm,this chapter introduces the bivariate difference field(F(α,β),σ),where α,β are two algebraically independent transcendental elements,σ is a transformation that satisfies σ(α)=β,σ(β)=uα+vβ,where u,v≠0∈F.Base on this consideration,the structure of such summations provides an algebraic framework for solving the difference equations of form aσ(g)+bg=f in the bivariate difference field(F(α,β),σ),where a,b,f∈F(α,β)are known binary functions of α,β.To solve the difference equations,this chapter firstly studies the basic properties of the bivariate difference filed and gives some results.Then it considers the polynomial solution and rational solution of the difference equations.To get the polynomial solution,it provides a total degree bound so that the polynomial solution could be solved directly by using the undetermined coefficients method.To give the rational solution,this chapter describes algorithms for finding the universal denominator for those equations which reduces the problem of finding the rational solutions to the problem of finding the polynomial solutions.