An Automatic Approach To The Generating Functions Of Some Special Sequences

In the thesis, algorithms for verifying some special combinatorial identities and computing the generating functions of some special sequences are studied. The main contents of this thesis can be summarized as follows:Chapter 1 describes the evolution of the proof theory of combinatorial identities.Chapter 2 is about the fundamental theory of the automatic approach to combinatorial identities, that is, holonomic functions and noncommutative operator algebras.Because the present Zeilberger algorithm requires a lot of memory so that some identities, such as Dixon's identity, can not be verified successfully, we generalize the Euclidean algorithm to the noncommutative operator algebra C in Chapter 3, and establish the elimination in the noncommutative context. By the use of the elimination instead of Sylvester's dialytic elimination, we modify the Zeilberger algorithm so that terminating hypergeometric identities (binomial coefficients identities) can be verified automatically. Besides, with the Maple program based on the modified algorithm, we have proven the identities in [26], including Saalschutz's identity, the Vandermonde-Chu convolution formula, and Dixon's identity, and 196 identities in [10]. Moreover, with the Wu method, we construct the way of eliminating several variables and develop an automatic approach to proving terminating hypergeometric identities of several variables. At the end of this chapter, we introduce the main results in C and discuss the problem of verifying the identities with the integral sign.Based on the discussion of Chapter 3, in Chapter 4, we consider the generating functions as special identities which contain continuous variables as well as discrete ones, and establish algorithms for computing the generating functions of some special sequences. In this chapter, we first give the algorithm for computing the ordinary power series generating functions and the exponential generating functions. By the use of this algorithm, we compute the generating functions of some orthogonal polynomials and some special combinatorial sequences. Next, we study the method of computing 2-variable ordinary power series generating functions and mixed-type generating functions with the approach to eliminating several variables in the noncommutative context developed in Chapter 3. And at last, we generalize our discussion to the generating functions with general forms.