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Recurrent Linear Operators

2013
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Complex Analysis and Operator Theory
*

We study the notion of

doi:10.1007/s11785-013-0348-9
fatcat:esfwgnkq2neqdiqp6e4np32jky
*recurrence*and some of its variations for*linear*operators acting on Banach spaces. ... We characterize*recurrence*for several classes of*linear*operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. ... Some examples and characterizations of*recurrence*for special classes of*linear*operators have appeared in [15] . ...##
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A Linear Recurrence System

1994
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Proceedings of the American Mathematical Society
*

We investigate a function which will be used to evaluate the

doi:10.2307/2161208
fatcat:5u6pprjjdzgsda5jwntfdn24yq
*linear*recursion ¡-i xt = û/.o + 5Za'JxJ for i = 1,2,3, ... , n J=x where the a¡j 's are arbitrary numbers. ... A*LINEAR**RECURRENCE*SYSTEM A. BLASIUS (Communicated by William W. Adams) Abstract . We look at a triangular system of n equations and reinvestigate a related function introduced by Chen and Kuck. ... Time and processor bounds for solving the*linear**recurrence*system are then obtained. Sameh and Brent [SB] later presented an alternate derivation of this algorithm. ...##
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A linear recurrence system

1994
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Proceedings of the American Mathematical Society
*

We investigate a function which will be used to evaluate the

doi:10.1090/s0002-9939-1994-1249870-5
fatcat:ddwrvn3bfbbrnpwds7uihc2ws4
*linear*recursion ¡-i xt = û/.o + 5Za'JxJ for i = 1,2,3, ... , n J=x where the a¡j 's are arbitrary numbers. ... A*LINEAR**RECURRENCE*SYSTEM A. BLASIUS (Communicated by William W. Adams) Abstract . We look at a triangular system of n equations and reinvestigate a related function introduced by Chen and Kuck. ... Time and processor bounds for solving the*linear**recurrence*system are then obtained. Sameh and Brent [SB] later presented an alternate derivation of this algorithm. ...##
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2731. Linear recurrence relations

1957
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Mathematical Gazette
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With a

doi:10.1017/s0025557200236164
fatcat:xs67r72v4fdqve7nhprstfqpqi
*linear**recurrence*relation of the kth order (having distinct roots) it is possible to express utn+r as a homogeneous polynomial of the tth degree in un+ku, un+k-2 ... Un. ... Corresponding coefficients in utn+r for five successive values of r satisfy the original*recurrence*relation. ...##
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Linear Recurrences via Probability

2015
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The American mathematical monthly
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The long run behavior of a

doi:10.4169/amer.math.monthly.122.04.386
fatcat:6glppt2hrzd57map3hykdg3rke
*linear**recurrence*is investigated using standard results from probability theory. ... In their conclusion to [1] the authors note that the analysis of*linear**recurrences*provides lovely examples for any*linear*algebra or computer algebra class. ... For 1 ≤ n ≤ m let x n = a n , while for n > m let x n = α m x n−1 + α m−1 x n−2 + · · · + α 1 x n−m . (1) This*recurrence*can be expressed using*linear*algebra as follows. ...##
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Solving linear recurrence equations

2011
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ACM SIGSAM Bulletin
*

As an example, for the

doi:10.1145/1940475.1940515
fatcat:rwlxurnnw5a5njvv2gkbvcpow4
*recurrence*relation satisfied by A099364 from the OEIS, our program produces the solution: ... 'Find Liouvillian' is not unique in terms of its purpose but, for second order*recurrence*relations, it is faster than prior algorithms. ... 'Find Liouvillian' is not unique in terms of its purpose but, for second order*recurrence*relations, it is faster than prior algorithms. ...##
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Summations of Linear Recurrent Sequences
[article]

2018
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arXiv
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pre-print

All powers of Fibonacci seem to follow this nice pattern that a

arXiv:1710.11074v3
fatcat:iv46pnxzj5dezfu6a3ak6cmjfu
*linear**recurrence*where the terms do not depend on n suffices, instead of in general for our technique, where the*recurrence*may need rational ... That is, by repeatedly applying the relation, we can write each a j+k as a*linear*combination: a j+k = D−1 m=0 c k,j,m a k+m , where for the j < D, we just let c k,j,m = 1 j = m 0 j = m . ...##
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Shortest Two-way Linear Recurrences
[article]

2010
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arXiv
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pre-print

We give a new and simpler algorithm to compute a shortest two-way

arXiv:0911.5459v3
fatcat:ggev2e4nmfdo7eq3vkippdytp4
*linear**recurrence*. ... Salagean proposed an algorithm to find such a shortest 'two-way'*recurrence*-- which may be longer than a*linear**recurrence*for $s$ of shortest length $\LC_n$. ... Thus minimal polynomials of s correspond to shortest*linear**recurrences*for s. ...##
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Linear Recurrences for Cylindrical Networks

2017
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International mathematics research notices
*

We prove a general theorem that gives a

doi:10.1093/imrn/rnx241
fatcat:mfb4unupcje6xifsiicykhlztm
*linear**recurrence*for tuples of paths in every cylindrical network. This can be seen as a cylindrical analog of the Lindstr\"om-Gessel-Viennot theorem. ...*Linear**recurrences*for tuples of paths In the previous section, we have established Theorem 2.1 that gave a*linear**recurrence*relation for single paths inÑ. ...*Linear**recurrences*for single paths In this section, we first describe some standard material on*linear**recurrences*in Section 3.1, then we define cylindrical networks and related notions rigorously in ...##
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Linear recurrences indexed by ℤ
[article]

2021
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arXiv
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pre-print

This note considers

arXiv:1906.04311v2
fatcat:dqfsiq2uo5fctnocd6hg3uerpa
*linear**recurrences*(also called*linear*difference equations) in unknowns indexed by the integers. ... We characterize a unique \emph{reduced}*linear**recurrence*with the same solutions as a given*linear**recurrence*, and construct a \emph{solution matrix} which parametrizes the space of solutions. ...*linear**recurrences*. ...##
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Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, I
[article]

2021
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arXiv
*
pre-print

This has been generalized for any Positive

arXiv:2001.08455v6
fatcat:4mo7p3esyvgwldfnjesyx6uoti
*Linear**Recurrence*Sequence (PLRS), which informally is a sequence satisfying a homogeneous*linear**recurrence*with a positive leading coefficient and non-negative ... We develop the Zeroing Algorithm, a powerful helper tool for analyzing the behavior of*linear**recurrence*sequences. ... We say a*recurrence*relation is a Positive*Linear**Recurrence*Relation (PLRR) if there are non-negative integers L, c 1 , . . . , c L such that H n+1 = c 1 H n + · · · + c L H n+1−L , (1.1) with L, c 1 ...##
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Continued Fractions and Linear Recurrences

1993
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Mathematics of Computation
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We prove that the numerators and denommators of the convergents to a real irrational number θ satisfy a

doi:10.2307/2152958
fatcat:hnkj5xs2sfc3bpx7zbcf5ch5ru
*linear**recurrence*with constant coefficients if and only if θ is a quadratic irrational The proof ... «)«>ο satisfies a*linear**recurrence*with constant integral coefficients. (b) =Φ (c) : The proof proceeds in two stages. ...*recurrence*with constant complex coefficients; (b) (q n ) n >o satisfies a*linear**recurrence*with constant complex coefficients, (c) (a"\>o is an ultimately penodic sequence; (d) θ is a quadratic irrational ...##
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Zeros of linear recurrence sequences

1999
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Manuscripta mathematica
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So in studying (1.1) we study the zeros of

doi:10.1007/s002290050136
fatcat:mdzoca5xazc2rb2oykncowok6e
*linear**recurrence*sequences. ... It is well known that f (0), f (1) , . . . is a*linear**recurrence*sequence of order (f ), which is "non-degenerate". ...##
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Linear recurrences of order two

1967
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Journal of the Australian Mathematical Society
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Let {/(»)} be a

doi:10.1017/s1446788700005139
fatcat:hfhex2ubpnetnlxwmi3wcolrrq
*linear**recurrence*of order two with companion polynomial G(y) = y z -ay-b eQ\y] whose roots and ratios of roots are not roots of unity. ... It has been proved that such an upper bound exists which is independent of a but the bound is dependent on the particular*linear**recurrence*. K. ... Introduction Let {/(«)} be a*linear**recurrence*of order two, i.e., (1) /(»+2) = af(n+l)+bf(n), a.beQ, for all positive integers n. Its companion polynomial is G(y) = y 2 -ay-b. ...##
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The Combinatorialization of Linear Recurrences

2011
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Electronic Journal of Combinatorics
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We provide two combinatorial proofs that

doi:10.37236/2008
fatcat:bpm3rzxisvbjnpa5ufscjgp7qq
*linear**recurrences*with constant coefficients have a closed form based on the roots of its characteristic equation. ... But this approach has not easily generalized to*linear**recurrences*with constant coefficients other than 1 nor for higher order*recurrences*. ... But better still, the weightings generalize to any*linear**recurrence*with constant coefficients. ...
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