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22
Solving Difference Equations in Finite Terms
 J. Symbolic Comput
, 1998
"... We define the notion of a liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of liouvillian sequences if and only if the Galois group of the equation is solvable. Using this we give a procedure to determine the liouvillian s ..."
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Cited by 50 (3 self)
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We define the notion of a liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of liouvillian sequences if and only if the Galois group of the equation is solvable. Using this we give a procedure to determine the liouvillian solutions of such a difference equation.
Solving parameterized linear difference equations in terms of indefinite nested sums and products
 J. Differ. Equations Appl
, 2002
"... The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. ..."
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Cited by 43 (31 self)
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The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. Keywords: symbolic summation, difference fields, ΠΣextensions, ΠΣfields AMS Subject Classification: 33FXX, 68W30, 12H10 1
Differential Galois Theory of Linear Difference Equations
 MATHEMATISCHE ANNALEN
, 2008
"... We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general r ..."
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Cited by 39 (13 self)
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We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of
Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations
 J. Pure Appl. Algebra
, 1998
"... In this paper the notion of finite singularities of difference operators is introduced, in order to adapt methods for differential equations to the case of recurrence equations. 1 Introduction The goal of this paper is to introduce the notion of finite singularities of difference equations (recurr ..."
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Cited by 35 (9 self)
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In this paper the notion of finite singularities of difference operators is introduced, in order to adapt methods for differential equations to the case of recurrence equations. 1 Introduction The goal of this paper is to introduce the notion of finite singularities of difference equations (recurrence equations) and to give some of their applications. So far only the singularity at infinity has been studied because that is the only point in P 1 (C) which is invariant under the shift operator , where (x) = x+1. Since x 7! x+1 does not leave the elements of C invariant, the finite singularities should be elements of C=ZZ instead of C . In the theory of differential operators, a singularity p is studied by considering differential operators over C((x \Gamma p)), or C((1=x)) if p = 1. The shift operator : C(x) ! C(x) can be extended to C((1=x)) but not to C((x \Gamma p)) for p 2 C , and so for finite singularities we can not use a construction similar to the differential case. For a...
Multibasic and Mixed Hypergeometric GosperType Algorithms
"... Introduction and notation Let F be a field of characteristic zero and ht n i 1 n=0 a sequence of elements from F which is eventually nonzero. Call t n : 1. hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (n)t n+1 = p 2 (n)t n for all n; 2. qhypergeometric or basic hyp ..."
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Cited by 15 (0 self)
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Introduction and notation Let F be a field of characteristic zero and ht n i 1 n=0 a sequence of elements from F which is eventually nonzero. Call t n : 1. hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (n)t n+1 = p 2 (n)t n for all n; 2. qhypergeometric or basic hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (q n )t n+1 = p 2 (q n<
Shiftless decomposition and polynomialtime rational summation
 In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ISSAC ’03
, 2003
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 7 (1 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
An Algorithm to Compute Liouvillian Solutions of Prime Order Linear DifferenceDifferential Equations
"... A normal form is given for integrable linear differencedifferential equations {σ(Y) = AY, δ(Y) = BY}, which is irreducible over C(x, t) and solvable in terms of liouvillian solutions. We refine this normal form and devise an algorithm to compute all liouvillian solutions of such kind of systems of ..."
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Cited by 5 (2 self)
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A normal form is given for integrable linear differencedifferential equations {σ(Y) = AY, δ(Y) = BY}, which is irreducible over C(x, t) and solvable in terms of liouvillian solutions. We refine this normal form and devise an algorithm to compute all liouvillian solutions of such kind of systems of prime order. Key words: linear differencedifferential equations, normal form, algorithm, liouvillian solutions.
Liouvillian Solutions of Linear DifferenceDifferential Equations
"... For a field k with an automorphism σ and a derivation δ, we introduce the notion of liouvillian solutions of linear differencedifferential systems {σ(Y) = AY, δ(Y) = BY} over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. In the forthcoming ..."
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Cited by 4 (0 self)
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For a field k with an automorphism σ and a derivation δ, we introduce the notion of liouvillian solutions of linear differencedifferential systems {σ(Y) = AY, δ(Y) = BY} over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. In the forthcoming paper, we will propose an algorithm for deciding if linear differencedifferential systems of prime order have liouvillian solutions. Key words: linear differencedifferential equations, Galois theory, liouvillian sequences.
Valuations of rational solutions of linear difference equations at irreducible polynomials,
 Adv. Appl. Math.,
, 2011
"... Abstract We discuss two algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, construct a finite set M of polynomials, irreducible in k [x], such that if the given equation After this for each p(x) ∈ M the algorithms compute a l ..."
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Abstract We discuss two algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, construct a finite set M of polynomials, irreducible in k [x], such that if the given equation After this for each p(x) ∈ M the algorithms compute a lower bound for val p(x) F (x), which is valid for any rational function solution F (x) of the initial equation. The algorithms are applicable to scalar linear equations of arbitrary orders as well as to linear systems of firstorder equations. The algorithms are based on a combination of renewed approaches used in earlier algorithms for finding a universal denominator (Abramov, Barkatou), and on a denominator bound (van Hoeij). A complexity analysis of the two proposed algorithms is presented.
Partial denominator bounds for partial linear difference equations
 In Proceedings of ISSAC’10
, 2010
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