Liouvillian solutions of linear difference–differential equations

Feng, Ruyong; Singer, Michael F.; Wu, Min

doi:10.1016/j.jsc.2009.09.001

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…Proof. Let K be the total quotient ring of a σδ-Picard-Vessiot extension R = k{Z, 1 det Z } δ of k corresponding to (5.1), and let G be its σδ-Galois group over k. Then S = k[Z, 1 det Z ] is a σ-Picard-Vessiot extension for this equation, and its total quotient ring F is a subring of K [11,Cor. 20].…”

Section: Hypertranscendencementioning

confidence: 99%

Galois groups for integrable and projectively integrable linear difference equations

Arreche¹,

Singer²

2017

Journal of Algebra

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Abstract. We consider first-order linear difference systems over C(x), with respect to a difference operator σ that is either a shift σ : x → x + 1, q-dilation σ : x → qx with q ∈ C × not a root of unity, or Mahler operator σ : x → x q with q ∈ Z ≥2 . Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.

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Section: Hypertranscendencementioning

confidence: 99%

Galois groups for integrable and projectively integrable linear difference equations

Arreche¹,

Singer²

2017

Journal of Algebra

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“…For algorithms to find Liouvillian solutions, see [13], [25], [8], [14], [15], [24], [19], [20]. Λ(a 0 , a 1 , . .…”

Section: Liouvillian Solutionsmentioning

confidence: 99%

Solving Linear Recurrence Equations with Polynomial Coefficients

Petkovšek

Zakrajšek

2013

Texts &Amp; Monographs in Symbolic Computation

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Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a first-order linear recurrence with constant coefficients, and a definite proper-hypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d'Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d'Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for state-of-the-art algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given P-recursive sequence, prove the salient closure properties of d'Alembertian sequences, and present an alternative proof of a recent result of Reutenauer's that Liouvillian sequences are precisely the interlacings of d'Alembertian ones.

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“…To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. For irreducible L, finding such τ n + cφ is equivalent to computing Liouvillian solutions, see [9,Lemma 4.1], [8,Prop. 3.1], or [5,Prop.…”

Section: Introductionmentioning

confidence: 99%

“…55]. Computing Liouvillian solutions is already solved in [9,11,2,8] where this problem is reduced to computing hypergeometric solutions of some other operatorL (or system, in [8]) which has n times more finite singularities (defined in [10]) than the original operator L. The problem is that computing a hypergeometric solution is done with a combinatorial algorithm [12,7] where the number of combinations depends exponentially on the number of singularities. We give a more direct approach, based on Theorems 2 and 3, that avoids introducingL and the corresponding increase in the number of singularities.…”

Section: Introductionmentioning

confidence: 99%