Definite Sums of Hypergeometric Terms and Limits of P-Recursive Sequences

TL;DR

This thesis advances symbolic computation by improving algorithms for hypergeometric sums and exploring the properties of D-finite numbers, including their relation to well-known mathematical constants.

Contribution

It introduces a faster, certificate-free algorithm for computing minimal telescopers for hypergeometric terms and characterizes D-finite numbers as values of D-finite functions at algebraic points.

Findings

The new algorithm outperforms Zeilberger's classical algorithm in experiments.

Order bounds for minimal telescopers are derived, sometimes improving existing bounds.

D-finite numbers over Gaussian rationals correspond to values of D-finite functions at algebraic points.

Abstract

The ubiquity of the class of D-finite functions and P-recursive sequences in symbolic computation is widely recognized. In this thesis, the presented work consists of two parts related to this class. In the first part, we generalize the reduction-based creative telescoping algorithms to the hypergeometric setting, which allows to deal with definite sums of hypergeometric terms more quickly. We first modify the Abramov-Petkovsek reduction, and then design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms based on the modified reduction. This new algorithm can avoid the costly computation of certificates, and outperforms the classical Zeilberger algorithm no matter whether certificates are computed or not according to the computational experiments. Moreover, we also derive order bounds for minimal telescopers. These bounds are sometimes better, and never worse than the known ones. In the second part of the thesis, we study the class of D-finite numbers. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D-finite numbers over the Gaussian rational field are essentially the same as the values of D-finite functions at non-singular algebraic number arguments (so-called the regular holonomic constants). This result makes it easier to recognize certain numbers as belonging to this class.