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We propose a simple algorithm to construct a polynomial divisible by the denominator of any rational solution of a linear difference equation an(z)y(z + n) + . . . + ao(x)y(z) = b(z) with polynomial coefficients and a polynomial right-hand side. Then we solve the same problem for q-difference equations. Nonhomogeneous equations with hypergeometric righthand… (More)
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=n0 R(k) = F (n) ∏n−1 k=n0 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)
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantages, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to… (More)
We present some algorithms related to rings of Ore polynomials (or, briefly, Ore rings) and describe a computer algebra library for basic operations in an arbitrary Ore ring. The library can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
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)
We propose a new algorithm for indefinite rational summation which, given a rational function F(z), extracts a rational part R(z) from the indefinite sum of F(z): If H(x) is not equal to O then the denominator of this rational function has the lowest possible degree. We then solve the same probleme in the g-difference case. 1 The decomposition problem We… (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>).
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. ©… (More)
Many computer algebra algorithms are based on constructing some polynomials and rational functions. Constructing a polynomial often involves finding the bound for its degree and use of the unknown coefficient method. For constructing rational functions cme expands the given functions in partial fractions and from these expansions obtains some hypothesis on… (More)