We describe a multiplicative normal form for rational functions which exhibits the shift structure of the factors, and investigate its properties. On the basis of this form we propose an algorithm which, given a rational function R, extracts a rational part F from the product of consecutive values of R: n−1 k=n 0 R(k) = F (n) n−1 k=n 0 V (k) where the… (More)
We describe a new direct algorithm for transforming a linear system of recurrences into an equivalent one with nonsingular leading or trailing matrix. Our algorithm, which is an improvement to the EG elimination method , uses only elementary linear algebra operations (ranks, kernels and determinants) to produce an equation satisfied by the degrees of the… (More)
A terminating condition of the well-known Zeilberger's algorithm for a given hypergeometric term <i>T</i>(<i>n, k</i>) is presented. It is shown that the only information on <i>T</i>(<i>n, k</i>) that one needs in order to determine in advance whether this algorithm will succeed is the rational function <i>T</i>(<i>n, k</i> + 1)/<i>T</i>(<i>n, k</i>).
1 Introduction Let K be a field of characteristic O and L : K[Z]-+ K[Z] an endomorphism of the K-linear space of univariate poly-nomials over K. We consider the following computational tasks concerning L: Tl, T2< T3. Homogeneous equation Ly = O: Compute a basis of Ker L in K[z]. Inhomogeneous equation Ly = f: Given ~ 6 K[z], compute a basis of the affine… (More)
We present an algorithm which, given a hypergeometric term <i>T</i>(<i>n</i>), constructs hypergeometric terms <i>T</i><subscrpt>1</subscrpt>(<i>n</i>) and <i>T</i><subscrpt>2</subscrpt>(<i>n</i>) such that <i>T</i>(<i>n</i>) = <i>T</i><subscrpt>1</subscrpt>(<i>n</i> + 1) -<i>T</i><subscrpt>1</subscrpt>(<i>n</i>) + <i>T</i><subscrpt>2</subscrpt>(<i>n</i>),… (More)
This paper is an exposition of different methods for computing closed forms of definite sums. The focus is on recently-developed results on computing closed forms of definite sums of hypergeometric terms. A design and an implementation of a software package which incorporates these methods into the computer algebra system Maple are described in detail.
Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.
We consider sequences which satisfy a linear recurrence equation Ly = 0 with polynomial coefficients. A criterion, i.e., a necessary and sufficient condition is proposed for validity of the discrete Newton-Leibniz formula when a primitive (an indefinite sum) Rt of a solution t of Ly = 0 is obtained either by Gosper's algorithm or by the Accurate Summation… (More)
D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reducing the order. Knowing d'Alembertian solutions of <italic>Ly =… (More)