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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 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 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 right-hand… (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 an algorithm to compute rational function solutions for a first order system of linear difference equations with rational coefficients. This algorithm does not require preliminary uncoupling of the given system.
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>).
We propose a new algorithm for indefinite rational summa-tion 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)
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.
Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. We prove a version of this conjecture in the case of two discrete variables.