Order Linear Recurrence Research Articles

Positivity of third order linear recurrence sequences

It is shown that the Positivity Problem for a sequence satisfying a third order linear recurrence with integer coefficients, i.e., the problem whether each element of this sequence is nonnegative, is decidable.

Rational approximations for values of derivatives of the Gamma function

The arithmetic nature of Euler’s constant $\gamma$ is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions $a_n$ and $b_n$ such that $a_n/b_n$ converges sub-exponentially to $\gamma$, viewed as $-\Gamma ’(1)$, where $\Gamma$ is the usual Gamma function. Although this is not yet enough to prove that $\gamma \not \in \mathbb {Q}$, it is a major step in this direction. In this paper, we present a different, but related, approach based on simultaneous Padé approximants to Euler’s functions, from which we construct and study a new third order recurrence that produces a sequence in $\mathbb {Q}(z)$ whose height grows like the factorial and that converges sub-exponentially to $\log (z)+\gamma$ for any complex number $z\in \mathbb {C}\setminus (-\infty ,0]$, where $\log$ is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to $\Gamma ^{(s)}(1)$ for any integer $s\ge 1$. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers $\gamma$ and $\Gamma ''(1)-2\Gamma ’(1)^2=\zeta (2)-\gamma ^2.$

Purely Periodic Second Order Linear Recurrences

Purely Periodic Second Order Linear Recurrences

On Sums Of Second Order Linear Recurrences by Hessenberg Matrices

On Sums Of Second Order Linear Recurrences by Hessenberg Matrices

Sums of the squares of terms of sequence {u n }

In this paper, we consider generalized Fibonacci type second order linear recurrence {un}. We derive a generating matrix for both the sums of squares, ∑i=0nui2 and the products of the form unun+2. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms unun+1 as well as the product, unun+2. Further we give generating functions and combinatorial representations of the sums of squares of terms of {un} and the product, unun+2.

HYERS-ULAM-RASSIAS STABILITY OF A SYSTEM OF FIRST ORDER LINEAR RECURRENCES

In this paper we discuss the Hyers-Ulam-Rassias stability of a system of first order linear recurrences with variable coefficients in Banach spaces. The concept of the Hyers-Ulam-Rassias stability originated from Th. M. Rassias# stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. As an application, the Hyers-Ulam-Rassias stability of a p-order linear recurrence with variable coefficients is proved.

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Irrationality measures are given for the values of the series $$\sum_{n=0}^{\infty} t^{n}/W_{an+b}$$ , where $$a,b\in\mathbb{Z}^+, 1\le b\le a, (a,b)=1$$ and W n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers $$ \sum_{n=0}^\infty \frac{1}{f_{an+b}},\quad \sum_{n=0}^\infty \frac{1}{l_{an+b}}, $$ where f n and l n are the Fibonacci and Lucas numbers, respectively.

Explicit bounds for multidimensional linear recurrences with restricted coefficients

This note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions. The results imply optimal zero-free polydisks for multivariable power series with 0, 1 coefficients. Among the applications is a result that states that the optimal zero-free polydisk has radius approximately 1 / ( v + 1 ) , for large dimensions v.

Hyers–Ulam–Rassias stability of a linear recurrence

In this paper we give a Hyers–Ulam–Rassias stability result for the first order linear recurrence in Banach spaces.

Parallel Processing of First Order Linear Recurrence on SMP Machines

In this paper, we propose a new algorithm that analyzes the data dependency pattern in the first-order linear recurrence (FOLR) and transforms it into algebraically equivalent expanded form so that it can be processed in parallel using the threads on symmetric multiprocessor (SMP) machines. The transformation aims to eliminate the data dependencies in the naive nested form of the FOLR. However, as this transformation may result in extra multiplication operations, our algorithm examines the immanent overhead of the expanded form of the FOLR and generates a new hybrid form of the FOLR. The hybrid form combines nested and appropriately expanded form in order to make it suitable for parallel processing. The parallel algorithm based on the hybrid form of the FOLR is analytically examined and tested through implementation on SMP machines. The implementation details, such as the workload balancing between processors and the optimization of cache performance, are also discussed. The experimental results show that the parallel algorithm based on the hybrid form of the FOLR considerably improves the performance on SMP machines that have three of more processors.

A cubic analogue of the RSA cryptosystem

In this paper, we investigate a public key cryptosystem which is derived from a third order linear recurrence relation and is analogous to the RSA and LUC cryptosystems. The explicit formulation involves a generalisation of the rule for composition of powers and of the calculus of the Euler totient function which underlie the algebra of the RSA cryptosystem. The security of all these systems appears to be comparable and to depend on the intractability of factorization but the systems do not seem to be mathematically equivalent.

Fourth Order Linear Recurrences Satisfied by Wythoff Pairs

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs $$(\left\lfloor {i\Phi } \right\rfloor ,\left\lfloor {i\Phi ^2 } \right\rfloor )$$ , where i is a natural number and φ is the golden ratio $$\frac{1}{2}(1 + \sqrt {5)} $$ . It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if φ is adjoined to the rationals.

Multiple orthogonal polynomials associated with macdonald functions

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bassel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation.

Counting divisors of Lucas numbers

Let {Sn} be a second order linear recurrence consisting of integers only. M. Ward [22] proved that, except for some degenerate cases, there are always an infinite number of distinct primes dividing the terms of {Sn}. A deeper question is whether in the non-degenerate case the set of prime divisors has a prime density. (If S is any set of natural numbers, then S(x) denotes the number of elements n in S with 1 < n ≤ x. In case S is a set of primes we define the prime density of S to be limx→∞ S(x)/π(x), if it exists, where π(x) denotes the number of primes not exceeding x.) It is conjectured that the answer is yes and that the density is in fact positive. In case of what are called torsion sequences, this was recently established by P. Stevenhagen [21], generalizing on results in the earlier papers [9, 11, 13]. Stevenhagen showed, moreover, that the density of a torsion sequence is a rational number. For a large class of non-torsion sequences, the existence and positivity of of the prime density was established by P.J. Stephens [20], under the assumption of the Generalized Riemann Hypothesis. The sequence {Ln} is torsion. Lagarias established that it has prime density 2/3. His method goes back to H. Hasse [6], who expressed the prime density of sequences {a+b}k=1 in terms of degrees of Kummer extensions. This method will be used in Section 3. The analytic aspects of prime divisors of sequences {ak + b}k=1 were explored by K. Wiertelak in several papers [24, 25, 26, 27, 28]. For a survey of results on prime divisors of, not necessarily second order, linear recurrences, see Ballot [1]. The problem of general divisors of second order linear recurrence sequences, in contrast, has not received much attention. Let a and b be fixed coprime integers such that |a| 6= |b|. In [12] the set of divisors, Ga,b, of the sequence {ak + bk} was considered. Some of the results obtained there have

Fourth Order Linearly Recurrent Wythoff Pairs

An examination of the sizable literature on Wythoff pairs, their generalizations, and Beatty sequences shows a considerable emphasis on connections with second order recurrence relations and quadratic irrationalities. However, it does not seem to have been previously noticed that a subsequence of the classical Wythoff pairs satisfies the irreducible fourth order linear recurrence $$C_{n + 4} = 10C_{n + 3} - 16C_{n + 2} + 5C_{n + 1} + C_n . $$ Thus, while Wythoff pairs are ordinarily associated with the roots of x2 − x − 1 = 0, there is a subset thereof that is associated with the roots of $$ x^4 - 10x^3 + 16x^2 - 5x - 1 = 0. $$

The distribution of second order linear recurrence sequences mod $2^m$

The distribution of second order linear recurrence sequences mod $2^m$

A parallel algorithm for evaluating general linear recurrence equations

A block parallel partitioning method for evaluating general linearmth order recurrence equations is presented and applied to solve the eigenvalues of symmetric tridiagonal matrices. The algorithm is based on partitioning, in a way that ensures load balance during computation. This method is applicable to both shared memory and distributed memory MIMD systems. The algorithm achieves a speedup ofO(p) on a parallel computer withp-fold parallelism, which is linear and is greater than the existing results, and the data communication between processors is less than that required for other methods. For solving symmetric tridiagonal eigenvalue problems, our method is faster than the best previous results. The results were tested and evaluated on an MIMD machine, and were within 79% to 96% of the predicted performance for the second order linear recurrence problem.

Comment: Remarks on LUC public key system

The authors of (Remarks on the LUC public key system, see ibid., vol. 30, p. 123, 1994) consider the Lucas function on which the public cryptosystem LUC proposed by Smith and Lennon (1993) is based. For a finite field G with m in G, the Lucas function V/sub n/(m) is defined by the second order linear recurrence relation V/sub n/(m)=mV/sub n-1/(m)-V/sub n-2/(m) where V/sub o/(m)=2, V/sub 1/(m)=m. The characteristic polynomial f of the Lucas function is f(x)=x/sup 2/-mx+1, and V/sub x/(m)= alpha /sup x/+ alpha /sup -x/ where alpha is a root of f. We now consider some of the results of and give concise proofs. >

A chained-matrices approach for parallel computation of continued fractions and its applications

A chained-matrices approach for parallel computing thenth convergent of continued fractions is presented. The resulting algorithm computes the entire prefix values of any continued fraction inO(logn) time on the EREW PRAM model or a network withO(n/logn) processors connected by the cube-connectedcycles, binary tree, perfect shuffle, or hypercube. It can be applied to approximate the transcendental numbers, such as π ande, inO(logm) time by usingO(m/logm) processors for a result withm-digit precision. We also use it to costoptimally solve the second-order linear recurrence, the polynomial evaluation, the recurrence of vector norm, the general class of recurrence equation defined by Kogge and Stone (1973), and the generalmth order linear recurrence. It is easy to implement because there are only some matrix multiplications and a division operation involved.

An O(log N) algorithm to solve linear recurrences on hypercubes

The hypercube is one of the most appealing and successful topologies among the interconnection networks designed for parallel processing applications. Linear recurrences arise frequently in many useful and important engineering and scientific applications. Consequently, fast parallel algorithms to solve linear recurrences offer significant value to the scientific community. In this paper we describe an efficient collision-free algorithm to solve general first order linear recurrences of length N on an N-node hypercube network. The time complexity of this algorithm is shown to be O(log N). For linear recurrences of length s × N where s is an integer > 0, the time complexity is shown to be O( s log N).