This thesis is mainly concerned with Gospers algorithm, its generalization, and rational solutions. Also it is concerned with the corresponding problems for the q-case. In Chapter 1, we first give a background on computerized proofs of identities, then we give some notations and definitions that are used throughout the entire thesis.After giving the basic results on Gospers algorithm, hypergeometric solutions of linear recurrences, and rational solutions of linear difference equations, we present the original approach of Gospers algorithm for indefinite hypergeometric summation together with the derivation of Gospers ansatz. In Chapter 2, we show that the uniqueness of the Gosper-Petkovsek representation of rational functions can be utilized to give a simpler version of Gospers algorithm. This approach also generalized to find hypergeometric solutions of linear recurrence equations with polynomial coefficients of any order, provided their leading and trailing coefficients are constant. Then we solve the same problems for the q-case. In Chapter 3, for given two polynomials, we discover a convergence property of the GCD of the rising factorial of a polynomial and the falling factorial of another polynomial, which serves as a simple approach to Gospers algorithm and the explicit formula for Abramovs universal denominator for linear difference equations with polynomial coefficients. We use this explicit formula to compute rational solutions of linear dif ference equations with polynomial coefficients and hypergeometric solutions of linear recurrence equations with polynomial coefticients, provided their leading and trailing coefficients are constant. In addition, we derive Abramovs universal denominator from Barkatous explicit formula. For a special case of first order linear difference equation we show that the universal denominators given by GP algorithm, and algorithm VMULT are factors of the universal denominator given by Abramovs algorithm and they are equal to each other. Then we show that our derivation of Gospers algorithm and the explicit formula for Abramovs universal denominator can be carried over to the q-case. To illustrate the applicability of our approaches, some examples are presented.Keywords: hypergeometric solution, Gospers algorithm, GP representation, rational solution, Abramovs algorithm, universal denominator, q-hypergeometric solution, qGospers algorithm, q-GP representation, q-rational solution.