scholarly journals Solution of a Linear Recurrence Equation Arising in the Analysis of Some Algorithms

1987 ◽  
Vol 8 (2) ◽  
pp. 233-250 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Recurrence Equation ◽  
Linear Recurrence

scholarly journals Generalized Alcuin's Sequence

10.37236/2796 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Panario ◽  
Murat Sahin ◽  
Qiang Wang
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Recurrence Equation ◽  
Linear Recurrence ◽  
Fixed Integer ◽  
New Family ◽  
Special Case

We introduce a new family of sequences $\{t_k(n)\}_{n=-\infty}^{\infty}$ for given positive integer $k$. We call these new sequences asgeneralized Alcuin's sequences because we get Alcuin's sequence which has several interesting properties when $k=3$. Also, $\{t_k(n)\}_{n=0}^{\infty}$ counts the number of partitions of $n-k$ with parts being $k, \left(k-1\right), 2\left(k-1\right),$ $3\left(k-1\right)$, $\ldots, \left(k-1\right)\left(k-1\right)$. We find an explicit linear recurrence equation and the generating function for $\{t_k(n)\}_{n=-\infty}^{\infty}$. For the special case $k=4$ and $k=5$, we get a simpler formula for $\{t_k(n)\}_{n=-\infty}^{\infty}$ and investigate the period of $\{t_k(n)\}_{n=-\infty}^{\infty}$ modulo a fixed integer. Also, we get a formula for $p_{5}\left(n\right)$ which is the number of partitions of $n$ into exactly $5$ parts.

Recurrence Contribution to Determining the Wear of Slide Bearings

2015 ◽  
Vol 220-221 ◽  
pp. 301-306
Author(s):  
Krzysztof Wierzcholski ◽  
Andrzej Miszczak
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Operating Time ◽  
Variable Coefficients ◽  
Recurrence Equation ◽  
Hydrodynamic Theory ◽  
Linear Recurrence ◽  
Recurrence Equations ◽  
Constant Coefficients

Numerous problems of numerical calculations occurring in power-train tribology and transport concerning the process of determining the problems of worn out bearings demand more and more information referring to the anticipation of the wear of slide bearings in the succeeding years of machine operations. Therefore, the paper presents the methods for working towards solutions to some specific classes of second and higher order non-homogeneous recurrence equations with variable coefficients occurring in the hydrodynamic theory of the problems dealing with bearing wear. Contrary to linear recurrence equations with constant coefficients, linear recurrence equations with variable coefficients rarely have analytical solutions. Numerical answers to such equations are always practicable. With reference to a large number of analytical methods for solutions to linear recurrence equations with variable coefficients, three research directions are usually followed.The first one depends upon the successive determination of particular linear independent solutions to the considered recurrence equation. The second direction can be characterized by a reduction in the order of the recurrence equation for obtaining the first order always solved recurrence equation. The third direction of solutions to recurrence equations with variable coefficients contains the methods for analytical solutions by means of a summation factor.The majority of the general methods of analytical solutions to linear recurrence equations with variable coefficients constitute the adaptation of the methods applied in solutions to suitable differential equations. As regards the final conclusions, the application of the theory presented in this paper contains numerical solutions referring to the wear values of the slide journal-bearing system in the indicated period of operating time.

scholarly journals Shift Equivalence of P-finite Sequences

10.37236/1126 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Manuel Kauers
Keyword(s):  
Equivalence Problem ◽  
Recurrence Equation ◽  
The Other ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Shift Equivalence ◽  
Finite Sequences ◽  
Homogeneous Linear

We present an algorithm which decides the shift equivalence problem for P-finite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shifting one of the sequences $s$ times makes it identical to the other, for some integer $s$. Our algorithm computes, for any two P-finite sequences, given via recurrence equation and initial values, all integers $s$ such that shifting the first sequence $s$ times yields the second.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals Period of a linear recurrence

1981 ◽  
Vol 39 (4) ◽  
pp. 303-311 ◽  
Author(s):  
A. Vince
Keyword(s):  
Linear Recurrence

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals Matrices determined by a linear recurrence relation among entries

2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation
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