A New Family of Somos-like Recurrences

Heideman, Paul; Hogan, Emilie

doi:10.37236/778

Cited by 9 publications

(30 citation statements)

References 3 publications

(7 reference statements)

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“…This agrees with the result of [14, Theorem 5.1], and with the particular integer sequence considered in [12]: putting the initial data x 0 = x 1 = x 2 = 1 with a = 1 into (3.16) gives the constant value K = 14, in accordance with Proposition 1.1 for k = 1.…”

Section: Linearizabilitysupporting

confidence: 91%

“…However, observe that the right-hand side of (1.4) has N terms, which means that for N 3 it cannot be obtained from an exchange relation in a cluster algebra, since for that to be so the polynomial F should have the same binomial form as (1.3); the case N = 2 is not an exchange relation either (but see [8,Example 4.11]). The inspiration for (1.4) comes from results of Heideman and Hogan [12], who considered odd order recurrences of the form 5) and showed that an integer sequence is generated for initial data…”

Section: Introductionmentioning

confidence: 99%

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A family of linearizable recurrences with the Laurent property

Hone

Ward

2014

Bulletin of the London Mathematical Society

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We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained.

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Section: Linearizabilitysupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

A family of linearizable recurrences with the Laurent property

Hone

Ward

2014

Bulletin of the London Mathematical Society

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“…Notice that it just depends upon the ℓ th column of the matrix. Since the matrix is skew-symmetric, the variable x ℓ does not occur on the right side of (17). After this process we have a new quiverQ, with a new matrixB.…”

Section: Recurrences With the Laurent Propertymentioning

confidence: 99%

Cluster mutation-periodic quivers and associated Laurent sequences

2010

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We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages, 35 figure

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“…which can be interpreted as a two-dimensional extension of (1). Another interesting linearizable mapping we mainly study in this paper is the Heideman-Hogan recurrence [14]:…”

Section: Introductionmentioning

confidence: 99%

A two-dimensional lattice equation as an extension of the Heideman–Hogan recurrence

Kamiya¹,

Kanki²,

Mase³

et al. 2018

J. Phys. A: Math. Theor.

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We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somoslike recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence.Higher order examples of two dimensional linearizable lattice equations related to the Dana-Scott recurrence are also discussed.

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A New Family of Somos-like Recurrences

Cited by 9 publications

References 3 publications

A family of linearizable recurrences with the Laurent property

A family of linearizable recurrences with the Laurent property

Cluster mutation-periodic quivers and associated Laurent sequences

A two-dimensional lattice equation as an extension of the Heideman–Hogan recurrence

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