Liouvillian solutions of irreducible second order linear difference equations

Hoeij, Mark van; Levy, Giles

doi:10.1145/1837934.1837991

Cited by 6 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Solutions of an equation of type 2 are called Liouvillian solutions [4,14,29]. Suppose u 1 is a solution of K = a 2 (x)τ 2 − a 0 (x) then {u 1 , u 2 }, where u 2 = (−1) x u 1 , forms a basis of V (K) and u 2 1 = u 2 2 .…”

Section: Symmetric Powers Of Operatorsmentioning

confidence: 99%

Closed form solutions of linear difference equations in terms of symmetric products

Cha

2014

Journal of Symbolic Computation

View full text Add to dashboard Cite

In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence equations and a counterpart to existing results on differential equations. As motivation and application for this work, we discuss the problem of proving positivity of sequences given merely in terms of their defining recurrence relation. The main advantage of the present approach to earlier methods attacking the same problem is that our algorithm provides human-readable and verifiable, i.e., certified proofs.

show abstract

Section: Symmetric Powers Of Operatorsmentioning

confidence: 99%

Closed form solutions of linear difference equations in terms of symmetric products

Cha

2014

Journal of Symbolic Computation

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…

The software to be presented is an implementation of the algorithms in [1], [2], and [3]. (This software is available at [4].)

…”

mentioning

confidence: 99%

Solving linear recurrence equations

Cha

Hoeij

Levy

2011

ACM Commun. Comput. Algebra

Self Cite

View full text Add to dashboard Cite

The software to be presented is an implementation of the algorithms in [1], [2], and [3]. (This software is available at [4].) The main algorithm in [3] is currently implemented with additional base equations beyond what appear in [3] and will be part of the software demonstration in July.Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations). Before calling any of these algorithms we first check currently available algorithms for a solution in the form of a first order right hand factor or some other general term solution.Algorithm 'Find 2 F 1 ' will find a gt-transformation to a recurrence relation satisfied by a hypergeometric series u(n) = 2 F 1 a+n b c z , if such a transformation exists. As an example, sequence A005572 = [1, 4, 17, 76, 354, 1704, 8421, . . . ] from the OEIS ([5]) represents the "Number of walks on cubic lattice starting and finishing on the xy plane and never going below it." A005572 has offset 0 (i.e. the first entry in the list is A005572(0)) and satisfies: 12(n + 1)A005572(n) − (20 + 8n)A005572(n + 1) + (n + 4)A005572(n + 2) = 0.The output from our program is:The algorithm 'Find Liouvillian' will find a gt-transformation to a recurrence relation of the form u(n + 2) + b(n)u(n) = 0 for some b(n) ∈ C(n), if such a transformation exists. 'Find Liouvillian' is not unique in terms of its purpose but, for second order recurrence relations, it is faster than prior algorithms. As an example, for the recurrence relation satisfied by A099364 from the OEIS, our program produces the solution:

show abstract

Liouvillian solutions of irreducible second order linear difference equations

Cited by 6 publications

References 11 publications

Closed form solutions of linear difference equations in terms of symmetric products

Closed form solutions of linear difference equations in terms of symmetric products

Solving Linear Recurrence Equations with Polynomial Coefficients

Solving linear recurrence equations

Contact Info

Product

Resources

About