Second-order Linear Recurrence Research Articles

Second-order linear recurrences with identically distributed residues modulo p^e

Let $p$ be an odd prime and let $u(a,-1)$ and $u(a',-1)$ be two Lucas sequences whose discriminants have the same nonzero quadratic character modulo $p$ and whose periods modulo $p$ are equal. We prove that there is then an integer $c$ such that for all $d\in\mathbb Z_p$, the frequency with which $d$ appears in a full period of $u(a,-1)\pmod p$ is the same frequency as $cd$ appears in $u(a',-1)\pmod p$. Here $u(a,b)$ satisfies the recursion relation $u_{n+2}=au_{n+1}+bu_n$ with initial terms $u_0=0$ and $u_1=1$. Similar results are obtained for the companion Lucas sequences $v(a,-1)$ and $v(a',-1)$. This paper extends analogous statements for Lucas sequences of the form $u(a,1)\pmod p$ given in a previous article. We further generalize our results by showing for a certain class of primes $p$ that if $e>1$, $b=\pm 1$, and $u(a,b)$ and $u(a',b)$ are Lucas sequences with the same period modulo $p$, then there exists an integer $c$ such that for all residues $d\pmod{p^e}$, the frequency with which $d$ appears in $u(a,b)\pmod{p^e}$ is the same frequency as $cd$ appears in $u(a',b)\pmod{p^e}$.

A unified approach to derive explicit solutions of generalized second-order linear recurrences and applications

We illustrate the versatility of our previous work on closed-form solutions of second-order linear recurrences by specializing the underlying counting sets to prove new identities. Extensions allowing the sequence values to be functions of a variable t as well as the integer variable n are mentioned and applied to derive closed-form solutions, both classical and new. Connections with continued fractions are established and used to settle several conjectures from the Ramanujan Machine. Finally, alternative verifications of well-known continued fractions such as those of zeta functions and continued fractions in Ramanujan’s second notebook are presented.

On classical orthogonal polynomials and the Cholesky factorization of a class of Hankel matrices

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.

Consecutive Ratios in Second-Order Linear Recurrence Sequences

Abstract Let (a n )n=0 ∞ be a second-order linear recurrence sequence with constant coefficient. We study the limit points and asymptotic distribution of the sequence of consecutive ratios a n +1/a n .

Generating functions of binary products of (p,q)-Fibonacci-like numbers with odd and even certain numbers and polynomials

In this paper, we study the symmetric and the generating functions for odd and even terms of the second-order linear recurrence sequences. we introduce a operator in order to derive a new family of generating functions of odd and even terms of Mersenne numbers, Mersenne Lucas numbers, (p,q)- Fibonacci-like numbers, k-Pell polynomials and k-Pell Lucas polynomials. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-Fibonacci-like numbers with odd and even terms of certain numbers and polynomials.

New Curiosity Bivariate Quadratic Quaternionic Polynomials and Their Roots

Abstract We consider the second-order linear homogeneous quaternion recurrence solutions for some new curiosity bivariate quadratic quaternionic equations.

A new class of generating functions of binary products of Gaussian numbers and polynomials

In this paper, we introduce a operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers and Gaussian Jacobsthal polynomials. In the paper Al4, Al5, a second-order linear recurrence sequence (U_{n}(a,b;p,q))_{n≥0} or briefly (U_{n})_{n≥0} is considered by the recurrence relation: U_{n+2}=pU_{n+1}+qU_{n}, with the initial conditions U₀=a and U₁=b, where a,b∈ℂ and p,q∈ℤ₊ for n≥0.

Horadam Sequences and Tridiagonal Determinants

We consider a family of particular tridiagonal matrix determinants which can represent the general second-order linear recurrence sequences. These determinants can be changed to symmetric or skew-symmetric tridiagonal determinants. To evaluate the complex factorizations of any Horadam sequence, we evaluate the eigenvalues of some special tridiagonal matrices and their corresponding eigenvectors. We also use these determinant representations to obtain some formulas in these sequences.

Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

On the ratios and geometric boundaries of complex Horadam sequences

Horadam sequences are second-order linear recurrences in the complex plane which depend on two initial conditions and two recurrence coefficients which are complex numbers. Recently, numerous papers have been devoted to the periodicity of these sequences, as well as to generalizations and applications. In this paper we investigate aspects related to the sequence of rations of consecutive terms and geometric bounds of Horadam sequences. We also propose some directions for further study.

S-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

Second-Order Recurrences with Nonconsecutive Initial Conditions

SummaryWhen consecutive initial conditions for second-order linear homogeneous recurrence relations with constant coefficients are given, the resulting sequence is uniquely determined. However, if the initial conditions are not consecutive, it may be the case that no sequence is possible, or that infinitely many sequences satisfy the recurrence.

Some identities related to Dedekind sums and the second-order linear recurrence polynomials

In this paper, we use the elementary method and the reciprocity theorem of Dedekind sums to study the computational problem of one kind Dedekind sums, and give two interesting computational formulae related to Dedekind sums and the second-order linear recurrence polynomials.

On the Smarandache-Pascal derived sequences and some of their conjectures

For any sequence , the Smarandache-Pascal derived sequence of is defined as , , , generally, for all , where is the combination number. In reference (Murthy and Ashbacher in Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences, 2005), authors proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence, one of them is that if , then we have the recurrence formula , . The main purpose of this paper is using the elementary method and the properties of the second-order linear recurrence sequence to study these problems and to prove a generalized conclusion.

On a continued fraction expansion for Eulerʼs constant

Recently, A.I. Aptekarev and his collaborators found a sequence of rational approximations to Eulerʼs constant γ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarevʼs approximations in terms of Meijer G -functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and γ with rational coefficients. Using this construction we find new rational approximations to γ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Eulerʼs constant. It seems to be the first non-trivial continued fraction expansion convergent to Eulerʼs constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler–Gompertz constant found by Stieltjes in 1895.

Computation of Private Key Based on Divide-By-Prime for Luc Cryptosystems

Problem statement: One of the public key cryptosystem is Luc cryptosystems. This system used Lucas Function for encryption and decryption process. Lucas Function is a special form of second-order linear recurrence relation. An encyption process is used to encrypt an original message to ciphertext by using public key. A decryption process is the process to decrypt a ciphertext into original message using private key. The existing algorithm on computing private key computation involved some redundant computations. Approach: In this study, an efficient algorithm to compute private key for Luc cryptosystem is developed. The Extended Euclidean Algorithm will be enhanced by implementing Divide-By-Prime in its computations. The comparison is focused on the computation time by the existing and new algorithms. The more efficient algorithm means the better computation time. The shorter computation time the better algorithm. Results: A new algorithm shows better computation time. In all experiments, the computation time by new algorithm is always better than the existing algorithm. Conclusion: The new computation algorithm that based on Divide-By-Prime provided better efficiency of decryption process compared to the existing algorithm.

Primitive prime factors in second-order linear recurrence sequences

For a class of Lucas sequences {xn}, we show that if n is a positive integer then xn has a primitive prime factor which divides xn to an odd power, except perhaps when n = 1, 2, 3 or 6. This has several desirable consequences.

Enumerating the Rationals from Left to Right

The rationals can be enumerated using Stern-Brocot sequences 풮ℬn, Calkin-Wilf sequences 풞풲n, or Farey sequences ℱn. We show that the rationals in 풮ℬn, 풞풲n, and ℱn can be computed using very similar second-order linear recurrence relations. (This, incidentally, obviates the need to compute previous sequences 풮ℬi, 풞풲i, ℱi with i < n.)

Reduce Computation Steps Can Increase the Efficiency of Computation Algorithm

Problem statement: Lucas function is a special form of second-order linear recurrence relation. It is used in LUC cryptosystems. The performance of LUC cryptosystem depends on the size of public key, messages and two relatively primes. The increasing of size of these parameters will increase the computation time need to perform the LUC Cryptosystem computation. The efficiency means the quality to avoid wasted time. Approach: Therefore, the main theme of this study was to design and implement an improve version of computation algorithm. The efficiency of computation can skip some computations time for a computerized calculation. Smaller computation time means the algorithm is better and more efficient than the other algorithm. In this study, the technique on reducing redundant number of computations steps in LUC Cryptosystem was investigated. The use of two variables w and t were proposed in order to reduce some computations steps in LUC Cryptosystem computation. Results: The new technique showed a better computation time compared to the existing algorithm. It also reduced some redundant multiplications without sacrificed the security of LUC Cryptosystem. At the same time, it increases the efficiency of computation algorithm. Conclusion: The proposed algorithm showed better speed and efficiency by reducing some redundant computation steps. It can reduce up to 20% of computation efforts compare to the existing one.

Recursive sequences and polynomial congruences

We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer m. We show that the period of such a sequence with characteristic polynomial f can be expressed in terms of the order of!D xCh fi as a unit in the quotient ring ZmT!UDZmTxU=h fi. When mD p is prime, this order can be described in terms of the factorization of f in the polynomial ring Z pTxU. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree k 5 in Z pTxU. This article grew out of an undergraduate research project, performed by the second author under the direction of the first, to determine if results about the periodicity of second-order linear homogeneous recurrence relations modulo positive integers could be extended to higher orders. We arrived, somewhat unexpectedly, at algorithms to determine the degrees of the irreducible factors of quintic and smaller degree polynomials modulo prime numbers. The algebraic properties of certain finite rings, particularly automorphisms of those rings, provided the connection between these two topics. To illustrate some of the ideas in this article, we begin with the famous example of the Fibonacci sequence, defined by FnD Fn 1C Fn 2 with F0D 0 and F1D 1. If, for some positive integer m, we replace each Fn by its remainder on division by m, we obtain a new sequence of integers. For example, the Fibonacci sequence modulo mD 10 begins 0; 1; 1; 2; 3; 5; 8; 3; 1; 4; 5; 9; 4; 3; 7; 0; 7; 7; 4; 1; 5; 6; 1; 7; 8; 5;:::; with the n-th term simply the last digit of Fn. We can also view such a sequence as having terms in ZmDZ=hmi, the ring of integers modulo m. This has the advantage