for finding more general solutions (e.g. rational or hypergeometric solutions) also rely on algorithms which compute polynomial solutions. Rational solutions In 1989 Sergei A. Abramov showed that a general rational solution, i.e. , with polynomial right-hand side , can be found by using the notion of a universal denominator. A universal denominator is a polynomial such that the denominator of every rational solution divides . Abramov showed how this universal denominator can be computed by only using the first and the last coefficient polynomial and . Substituting this universal denominator for the unknown denominator of all rational solutions can be found by computing all polynomial solutions of a transformed equation. Hypergeometric solution A sequence is called hypergeometric if the ratio of two consecutive terms is a rational function in , i.e. . This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients. The set of hypergeometric sequences is not a subspace of the space of sequences as it is not closed under addition. In 1992 Marko Petkovšek gave an algorithm to get the general hypergeometric solution of a recurrence equation where the right-hand side is the sum of hypergeometric sequences. The algorithm makes use of the Gosper-Petkovšek normal-form of a rational function. With this specific representation it is again sufficient to consider polynomial solutions of a transformed equation. A different and more efficient approach is due to Mark van Hoeij. Considering the roots of the first and the last coefficient polynomial and – called singularities – one can build a solution step by step making use of the fact that every hypergeometric sequence has a representation of the formfor some with for and . Here denotes the Gamma function and the algebraic closure of the field . Then the have to be singularities of the equation (i.e. roots of or ). Furthermore one can compute bounds for the exponents . For fixed values it is possible to make an ansatz which gives candidates for . For a specific one can again make an ansatz to get the rational function by Abramov's algorithm. Considering all possibilities one gets the general solution of the recurrence equation. D'Alembertian solutions A sequence is called d'Alembertian if for some hypergeometric sequences and means that where denotes the difference operator, i.e. . This is the case if and only if there are first-order linear recurrence