Some well-known correspondences between sets of linearly independent rows and columns of matrices over fields carry over to matrices over non-commutative rings without nontrivial zero divisors.

We propose four multiplicative canonical forms that exhibit the shift structure of a given rational function. These forms in particular allow one to represent a hypergeometric term efficiently. Each of these representations is optimal in some sense.

We present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) = T1(n + 1) -T1(n) + T2(n), and T2(n) is minimal in some sense. This solves the decomposition problem for indefinite sums of hypergeometric terms: T1(n + 1) - T1(n) is the “summable part” and T2(n) the “non-summable part” of T(n).

Introduction Let K be a field of characteristic 0 and L : K[x] ! K[x] an endomorphism of the K-linear space of univariate polynomials over K. We consider the following computational tasks concerning L: T1. Homogeneous equation Ly = 0: Compute a basis of Ker L in K[x]. T2. Inhomogeneous equation Ly = f : Given f 2 K[x], compute a basis of the affine space L \\Gamma1 (f) in K[x]. T3. Parametric inhomogeneous equation Ly = P m i=1 i f i : Given f1

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 Ly = 0, one can write down the corresponding solutions of Ly = f and of L*y = 0.

Hypergeometric sequences are such that the quotient of two successive terms is a fixed rational function of the index. We give a generalization of M. Petkovsek's algorithm to find all hypergeometric sequence solutions of linear recurrences, and we describe a program to find all hypergeometric functions that solve a linear differential equation. Solutions hyperg'eom'etriques des 'equations diff'erentielles lin'eaires R'esum'e Les suites hyperg'eom'etriques sont telles que le quotient de deux termes cons'ecutifs est une fonction rationnelle fixe de l'indice. Nous donnons une g'en'eralisation de l'algorithme de M. Petkovsek qui d'etermine toutes les solutions hyperg'eom'etriques de r'ecurrences lin'eaires, et nous d'ecrivons un programme qui donne toutes les fonctions hyperg'eom'etriques solutions d"equations diff'erentielles lin'eaires. To appear in Proceedings ISSAC'93. M. Bronstein ed. ACM Press. Finding All Hypergeometric Solutions of Linear Differential Equations Marko Petkovsek ...

The paper shows that certain interesting properties of roundings and representations hold at a fairly general level. The author restricts his attention to roundings of real numbers. He defines roundings by means of two parameters for each basic interval of the screen. These parameters determine the dividing point of the interval and the direction in which the dividing point moves under rounding. He shows that radix systems in which truncation and lexicographic ordering obey natural monotonicity conditions can be characterized as systems with contiguous digit sets. An examination is made of the interplay of roundings and representations.>