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Recurrent Linear Operators

George Costakis, Antonios Manoussos, Ioannis Parissis
2013 Complex Analysis and Operator Theory  
We study the notion of 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] .  ... 
doi:10.1007/s11785-013-0348-9 fatcat:esfwgnkq2neqdiqp6e4np32jky

A Linear Recurrence System

A. Blasius
1994 Proceedings of the American Mathematical Society  
We investigate a function which will be used to evaluate the 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.  ... 
doi:10.2307/2161208 fatcat:5u6pprjjdzgsda5jwntfdn24yq

A linear recurrence system

A. Blasius
1994 Proceedings of the American Mathematical Society  
We investigate a function which will be used to evaluate the 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.  ... 
doi:10.1090/s0002-9939-1994-1249870-5 fatcat:ddwrvn3bfbbrnpwds7uihc2ws4

2731. Linear recurrence relations

R. C. Lyness
1957 Mathematical Gazette  
With a 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.  ... 
doi:10.1017/s0025557200236164 fatcat:xs67r72v4fdqve7nhprstfqpqi

Linear Recurrences via Probability

Byron Schmuland
2015 The American mathematical monthly  
The long run behavior of a 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.  ... 
doi:10.4169/amer.math.monthly.122.04.386 fatcat:6glppt2hrzd57map3hykdg3rke

Solving linear recurrence equations

Yongjae Cha, Mark van Hoeij, Giles Levy
2011 ACM SIGSAM Bulletin  
As an example, for the 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.  ... 
doi:10.1145/1940475.1940515 fatcat:rwlxurnnw5a5njvv2gkbvcpow4

Summations of Linear Recurrent Sequences [article]

Andrew Lohr
2018 arXiv   pre-print
All powers of Fibonacci seem to follow this nice pattern that a 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 .  ... 
arXiv:1710.11074v3 fatcat:iv46pnxzj5dezfu6a3ak6cmjfu

Shortest Two-way Linear Recurrences [article]

Graham H. Norton
2010 arXiv   pre-print
We give a new and simpler algorithm to compute a shortest two-way 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.  ... 
arXiv:0911.5459v3 fatcat:ggev2e4nmfdo7eq3vkippdytp4

Linear Recurrences for Cylindrical Networks

Pavel Galashin, Pavlo Pylyavskyy
2017 International mathematics research notices  
We prove a general theorem that gives a 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  ... 
doi:10.1093/imrn/rnx241 fatcat:mfb4unupcje6xifsiicykhlztm

Linear recurrences indexed by ℤ [article]

Greg Muller
2021 arXiv   pre-print
This note considers 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.  ... 
arXiv:1906.04311v2 fatcat:dqfsiq2uo5fctnocd6hg3uerpa

Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, I [article]

Thomas C. Martinez, Steven J. Miller, Clayton Mizgerd, Chenyang Sun
2021 arXiv   pre-print
This has been generalized for any Positive 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  ... 
arXiv:2001.08455v6 fatcat:4mo7p3esyvgwldfnjesyx6uoti

Continued Fractions and Linear Recurrences

H. W. Lenstra, J. O. Shallit
1993 Mathematics of Computation  
We prove that the numerators and denommators of the convergents to a real irrational number θ satisfy a 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  ... 
doi:10.2307/2152958 fatcat:hnkj5xs2sfc3bpx7zbcf5ch5ru

Zeros of linear recurrence sequences

Hans Peter Schlickewei, Wolfgang M. Schmidt, Michel Waldschmidt
1999 Manuscripta mathematica  
So in studying (1.1) we study the zeros of linear recurrence sequences.  ...  It is well known that f (0), f (1) , . . . is a linear recurrence sequence of order (f ), which is "non-degenerate".  ... 
doi:10.1007/s002290050136 fatcat:mdzoca5xazc2rb2oykncowok6e

Linear recurrences of order two

R. R. Laxton
1967 Journal of the Australian Mathematical Society  
Let {/(»)} be a 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.  ... 
doi:10.1017/s1446788700005139 fatcat:hfhex2ubpnetnlxwmi3wcolrrq

The Combinatorialization of Linear Recurrences

Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn
2011 Electronic Journal of Combinatorics  
We provide two combinatorial proofs that 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.  ... 
doi:10.37236/2008 fatcat:bpm3rzxisvbjnpa5ufscjgp7qq
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