A NOTE ON NEW GENERALIZATIONS OF k-HORADAM SEQUENCES AND THE POWER SEQUENCES OF THESE GENERALIZATIONS

2021 ◽  
Vol 21 (3) ◽  
pp. 625-638
Author(s):  
CAGLA CELEMOGLU
Keyword(s):  
Number Theory ◽  
Generating Function ◽  
Coding Theory ◽  
Binet Formula ◽  
Power Sequence

In this article, firstly, we have described new generalizations of generalized k - Horadam sequence and we named the generalizations as another generalized k - Horadam sequence {H k,n}nE, a different generalized k - Horadam sequence {qk,n} and an altered generalized k - Horadam sequence {Qk,n) , respectively. Then, we have studied properties of these new generalizations and we have obtained generating function and extended Binet formula for each generalization. Also, we have introduced a power sequence for an altered generalized k - Horadam sequence in order to be used in different applications like number theory, cryptography, coding theory and engineering.

scholarly journals ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE COMPOSITIONS

2019 ◽  
Vol 100 (2) ◽  
pp. 230-238 ◽  
Author(s):  
NIAN HONG ZHOU
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Generating Function ◽  
Rank Statistic ◽  
Integer Partition

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

STURMIAN WORDS AND AMBIGUOUS CONTEXT-FREE LANGUAGES

1990 ◽  
Vol 01 (03) ◽  
pp. 309-323 ◽  
Author(s):  
FILIPPO MIGNOSI
Keyword(s):  
Number Theory ◽  
Generating Function ◽  
Rational Number ◽  
Asymptotic Density ◽  
Gap Theorem ◽  
Context Free Language ◽  
Sturmian Words ◽  
Context Free ◽  
Free Language

If x is a rational number, 0<x≤1, then A(x)c is a context-free language, where A(x) is the set of factors of the infinite Sturmian words with asymptotic density of 1’s smaller than or equal to x. We also prove a “gap” theorem i.e. A(x) can never be an unambiguous co-context-free language. The “gap” theorem is established by proving that the counting generating function of A(x) is transcendental. We show some links between Sturmian words, combinatorics and number theory.

scholarly journals A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

2020 ◽  
pp. 1-13 ◽  
Author(s):  
Sukran Uygun ◽  
Evans Owusu
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Convergence Property ◽  
Lucas Numbers ◽  
Lucas Sequence ◽  
Summation Formulas ◽  
Binet Formula

In this study, we bring into light a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as   with initial conditions $$\ \hat{c}_{0}=2,\ \hat{c}_{1}=a.$$ The Binet formula as well as the generating function for this sequence are given. The convergence property of the consecutive terms of this sequence is examined after which the well known Cassini, Catalan and the D'ocagne identities as well as some related summation formulas are also given.

scholarly journals Generalized Gaussian Jacobsthal and Generalized Gaussian Jacobsthal Lucas Polynomial

2019 ◽  
Vol 8 (6S3) ◽  
pp. 708-712
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binet Formula ◽  
Q Matrix

The aim of the present paper is to present a generalization of Gaussian Jacobsthal polynomial and Gaussian Jacobsthal Lucas polynomial. Present paper extends the work of Asci and Gurel [3]. Some important generalizations of Generating function, Binet formula, Explicit formula, Q matrix and determinantal representations of these polynomials are also produced.

scholarly journals ON THE BINOMIAL TRANSFORMS OF THE HORADAM QUATERNION SEQUENCES

2020 ◽  
Author(s):  
Faruk Kaplan ◽  
Arzu Özkoç Öztürk
Keyword(s):  
Recurrence Relation ◽  
General Formula ◽  
Generating Function ◽  
Second Order ◽  
Binet Formula ◽  
New Type ◽  
Binomial Transform

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.

scholarly journals Matrix Representation of Bi-Periodic Jacobsthal Sequence

2020 ◽  
pp. 1-12
Author(s):  
Sukran Uygun ◽  
Evans Owusu
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Matrix Representation ◽  
General Term ◽  
Identity Matrix ◽  
Summation Formulas ◽  
Matrix Sequence ◽  
Binet Formula ◽  
The Matrix ◽  
Generalized Matrix

In this paper, we bring into light the matrix representation of bi-periodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal matrix sequence. We dene it as with initial conditions J0 = I identity matrix, . We obtained the nth general term of this new matrix sequence. By studying the properties of this new matrix sequence, the well-known Cassini or Simpson's formula was obtained. We then proceed to find its generating function as well as the Binet formula. Some new properties and two summation formulas for this new generalized matrix sequence were also given.

scholarly journals On the permanent of Schur's matrix

1976 ◽  
Vol 21 (4) ◽  
pp. 487-497 ◽  
Author(s):  
R. L. Graham ◽  
D. H. Lehmer
Keyword(s):  
Number Theory ◽  
Coding Theory ◽  
Homogeneous Polynomial ◽  
Circulant Matrix ◽  
Image Position

AbstractSchur's matrix Mnis ordinarily defined to be thenbynmatrix (εjk), 0 ≦j, k < n, where ε = exp (2 πi/n). This matrix occurs in a variety of areas including number theory, statistics, coding theory and combinatorics. In this paper, we investigatePn, thepermanentofMn, which is define bywhere π ranges over alln! permutations on {0,1, …,n– 1}.Pnoccurs, for example, in the study of circulants. Specifically, letXndenote thenbyncirculant matrix(xi, j)withxi, j= xi, j, where the subscript is reduced modulon. The determinant ofXnis a homogeneous polynomial of degreenin thexiand can be written asThenPn= A(1,1, … 1). Typical of the results established in this note are: (i)P2n= 0 for alln, (ii)Pp≡p! (modp3)forpa prime >3. (iii) IfpadividesnthendividesPn. Also, a table of values ofPnis given for 1 ≦n≦ 23.

scholarly journals On split r-Jacobsthal quaternions

2020 ◽  
Vol 74 (1) ◽  
pp. 1
Author(s):  
Dorota Bród
Keyword(s):  
Generating Function ◽  
Binet Formula

In this paper we introduce a one-parameter generalization of the split Jacobsthal quaternions, namely the split r-Jacobsthal quaternions. We give a generating function, Binet formula for these numbers. Moreover, we obtain some identities, among others Catalan, Cassini identities and convolution identity for the split r-Jacobsthal quaternions.

Coding Theory and Number Theory

2003 ◽  
Author(s):  
Toyokazu Hiramatsu ◽  
Günter Köhler
Keyword(s):  
Number Theory ◽  
Coding Theory

scholarly journals Perrin’s bivariate and complex polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 70-78
Author(s):  
Renata Passos Machado Vieira ◽  
Milena Carolina dos Santos Mangueira ◽  
Francisco Regis Vieira Alves ◽  
Paula Maria Machado Cruz Catarino
Keyword(s):  
Generating Function ◽  
Complex Polynomials ◽  
Bivariate Polynomials ◽  
Binet Formula ◽  
Generating Matrix ◽  
Definition Of

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

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