scholarly journals On the Combinatorial Properties of Bihyperbolic Balancing Numbers

2020 ◽  
Vol 77 (1) ◽  
pp. 27-38
Author(s):  
Dorota Bród ◽  
Anetta Szynal-Liana ◽  
Iwona Włoch
Keyword(s):  
Generating Function ◽  
Combinatorial Properties ◽  
Balancing Numbers ◽  
Binet Formula

AbstractIn this paper, we introduce bihyperbolic balancing and Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities and the generating function.

scholarly journals On Some Combinatorial Properties of P (r, n)-Pell Quaternions

2020 ◽  
Vol 77 (1) ◽  
pp. 1-12
Author(s):  
Dorota Bród ◽  
Anetta Szynal-Liana
Keyword(s):  
Generating Function ◽  
Combinatorial Properties ◽  
Binet Formula

AbstractIn this paper we introduce a new one parameter generalization of the Pell quaternions – P (r, n)-Pell quaternions. We give some of their properties, among others the Binet formula, convolution identity and the generating function.

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

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 Distinct Parts Partitions without Sequences

10.37236/4970 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Karl Mahlburg ◽  
Karthik Nataraj
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Series Representation ◽  
Double Series ◽  
Asymptotic Formulas ◽  
Similar Investigation ◽  
Combinatorial Properties ◽  
Consecutive Integers

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.

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 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.

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.

scholarly journals SOME PROPERTIES OF BIVARIATE FIBONACCI AND LUCAS QUATERNION POLYNOMIALS

2020 ◽  
Vol 35 (1) ◽  
pp. 073
Author(s):  
Arzu Özkoç Öztürk ◽  
Faruk Kaplan
Keyword(s):  
Generating Function ◽  
Matrix Representation ◽  
Binet Formula ◽  
Fibonacci Quaternion

In this work, we introduce bivariate Fibonacci quaternion polynomials andbivariate Lucas quaternion polynomials. We present generating function,Binet formula, matrix representation, binomial formulas and some basicidentities for the bivariate Fibonacci and Lucas quaternion polynomialsequences. Moreover we give various kinds of sums for these quaternionpolynomials.

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