On linear recurrence sequences with polynomial coefficients

Poorten, Alfred J. van der; Shparlinski, Igor E.

doi:10.1017/s0017089500031372

Cited by 10 publications

(9 citation statements)

References 12 publications

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“…We write the "little-c" recursion in this way for convenient connection with experimental results; for example, we have always encountered natural alternating signs, and some obvious factors of the polynomials p implicitly defined by (37). Note, for instance, that the experimental recursions (35) and (36) can be recast compactly in the form of (37) by defining Table 2 has many other p n,i polynomials that we have found experimentally.…”

Section: Recursion Relations-experimentsmentioning

confidence: 99%

“…Most authors abide by the nomenclature as we do, namely, the order of the recursion is M , meaning there are M + 1 different C terms (and M + 1 polynomial coefficients). Some researchers refer to any sequence such as C, satisfying such a recursion, as holonomic, and observe that a generating function will satisfy a similar recurrence relation in its derivatives [37,47,20].…”

Section: Recursion Relations-experimentsmentioning

confidence: 99%

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Hypergeometric Forms for Ising-Class Integrals

Bailey

Borwein

2007

Experimental Mathematics

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We apply experimental-mathematical principles to analyze integralsThese are generalizations of a previous integral C n := C n,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the C n,k in terms of Meijer G-functions and nested-Barnes integrals. Our investigations began by computing 500-digit numerical values of C n,k for all integers n, k where n ∈ [2, 12] and k ∈ [0, 25]. We found that some C n,k enjoy exact evaluations involving Dirichlet L-functions or the Riemann zeta function. In the process of analyzing hypergeometric representations, we found-experimentally and strikingly-that the C n,k almost certainly satisfy certain inter-indicial relations including discrete k-recursions. Using generating functions, differential theory, complex analysis, and Wilf-Zeilberger algorithms we are able to prove some central cases of these relations.

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Section: Recursion Relations-experimentsmentioning

confidence: 99%

Section: Recursion Relations-experimentsmentioning

confidence: 99%

Hypergeometric Forms for Ising-Class Integrals

Bailey

Borwein

2007

Experimental Mathematics

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“…In fact, a slightly more precise version of Theorem 1.1 was proved by van der Poorten-Shparlinski [vdPS96,. Their method uses a technical construction of a certain auxiliary function.…”

Section: Introductionmentioning

confidence: 99%

D-finiteness, rationality, and height

Bell¹,

Nguyen²,

Zannier³

2020

Trans. Amer. Math. Soc.

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Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic 0 is D-finite and its coefficients belong to a finite set then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series "look like" the coefficients of a rational function then the power series is rational. Our work relies on the theory of Weil heights, the Manin-Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.Pisot's Hadamard Quotient Conjecture solved by Pourchet [Pou79] and van der Poorten [vdP88] (see Rumely's note [Rum88] for more details and the paper by Corvaja-Zannier [CZ02a] for a stronger version). Note that P 1 in the above results is the property that the given power series is a rational function. In this paper, we are interested in the situation when P 1 is the so called D-finiteness property.

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“…Heights of coefficients of D-finite power series have been studied independently, notably by van der Poorten and Shparlinski [vdPS96], who showed a gap result holds in this context that is somewhat weaker than what is predicted by our height gap conjecture above; specifically, they showed that if n≥0 a(n…”

Section: Introductionmentioning

confidence: 90%

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

Bell¹,

Ghioca²,

Satriano³

2019

Preprint

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We prove a uniform version of the Dynamical Mordell-Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our first result, we assume X is a quasi-projective variety defined over a field K of characteristic 0, endowed with the action of an étale endomorphism Φ, and f : X −→ Y is a morphism with Y a quasi-projective variety defined over K. Then for any x ∈ X(K), if for each y ∈ Y (K), the set Sy := {n ∈ N : f (Φ n (x)) = y} is finite, then there exists a positive integer N such that #Sy ≤ N for each y ∈ Y (K). For our second result, we let K be a number field, f : X P 1 is a rational map, and Φ is an arbitrary endomorphism of X. If OΦ(x) denotes the forward orbit of x under the action of Φ, then either f (OΦ(x)) is finite, or lim sup n→∞ h(f (Φ n (x)))/ log(n) > 0, where h(•) represents the usual logarithmic Weil height for algebraic points.

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On linear recurrence sequences with polynomial coefficients

Cited by 10 publications

References 12 publications

Hypergeometric Forms for Ising-Class Integrals

Hypergeometric Forms for Ising-Class Integrals

D-finiteness, rationality, and height

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

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