The ubiquity of the class of D-finite functions and P-recursive sequences in symbolic computation is widely recognized. This class is defined in terms of linear differential and difference equations with polynomial coefficients. 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. This generalization allows to deal with definite sums of hypergeometric terms more quickly. The Abramov-Petkovsek reduction computes an additive decomposition of a given hypergeometric term, which extends the functionality of Gosper's algorithm for indefinite hypergeometric summation. We modify this reduction so as to decompose a hypergeometric term as the sum of a summable term and a nonsummable one. Properties satisfied by the output of the original reduction carry over to our modified version. Moreover, the modified reduction does not solve any auxiliary linear difference equation explicitly. Based on the modified reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. 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. We further employ a new argument for the termination of the above new algorithm, which enables us to derive order bounds for minimal telescopers. Compared to the known bounds in the literature, our 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, which is closely related to D-finite functions and P-recursive sequences. 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.