scholarly journals Gaussian Pell and Gaussian Pell-Lucas quaternions

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1609-1617
Author(s):  
Hasan Arslan
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Summation Formulas

The main aim of this work is to introduce the Gaussian Pell quaternion QGpn and Gaussian Pell-Lucas quaternion QGqn, where the components of QGpn and QGqn are Pell numbers pn and Pell-Lucas numbers qn, respectively. Firstly, we obtain the recurrence relations and Binet formulas for QGpn and QGqn. We use Binet formulas to prove Cassini?s identity for these quaternions. Furthermore, we give some basic identities for QGpn and QGqn such as some summation formulas, the terms with negative indices and the generating functions for these complex quaternions.

scholarly journals Properties of hyperbolic generalized Pell numbers

2020 ◽  
Vol 26 (4) ◽  
pp. 136-153
Author(s):  
Yüksel Soykan ◽  
◽  
Melih Göcen ◽  
Keyword(s):  
Clifford Algebra ◽  
Generating Functions ◽  
Hyperbolic Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized hyperbolic Pell numbers over the bidimensional Clifford algebra of hyperbolic numbers. As special cases, we deal with hyperbolic Pell and hyperbolic Pell–Lucas numbers. We present Binet’s formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan’s, Cassini’s, d’Ocagne’s, Gelin–Cesàro’s, Melham’s identities and present matrices related to these sequences.

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals On Dual Hyperbolic Generalized Fibonacci Numbers

2019 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

scholarly journals On Generalized Third-Order Pell Numbers

2019 ◽  
pp. 1-18
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Third Order ◽  
Pell Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we investigate the generalized third order Pell sequences and we deal with, in detail, three special cases which we call them third order Pell, third order Pell-Lucas and modified third order Pell sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

scholarly journals Hyperbolic Jacobsthal Numbers

2019 ◽  
pp. 1-9
Author(s):  
Can Murat Dikmen
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Inner Product ◽  
Summation Formulas

In this paper, we introduce the Hyperbolic Jacobsthal numbers and we present recurrence relations, Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we investgate Lorentzian inner product for the hyperbolic Jacobsthal vectors.

scholarly journals Some properties of the generalized Fibonacci and Lucas numbers

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2655-2665
Author(s):  
Gospava Djordjevic ◽  
Snezana Djordjevic
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Fibonacci And Lucas Numbers

In this paper we consider the generalized Fibonacci numbers Fn,m and the generalized Lucas numbers Ln,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.

scholarly journals A Parametric Kind of the Degenerate Fubini Numbers and Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 405 ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Cheon Seoung Ryoo
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Summation Formulas ◽  
Degenerate Type

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling numbers.

scholarly journals ORDINARY GENERATING FUNCTIONS OF BINARY PRODUCTS OF (p,q)-MODIFIED PELL NUMBERS AND k-NUMBERS AT POSITIVE AND NEGATIVE INDICES

2020 ◽  
Vol 20 (3) ◽  
pp. 627-648
Author(s):  
NABIHA SABA ◽  
ALI BOUSSAYOUD
Keyword(s):  
Generating Functions ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Balancing Numbers ◽  
Symmetric Properties

In this paper, we introduce a operator in order to derive some new symmetric properties of (p,q)-modified Pell numbers and we give some new generating functions of the products of (p,q)-modified Pell numbers with k-Fibonacci and k-Lucas numbers, k-Pell and k-Pell Lucas numbers, k-Jacobsthal and k-Jacobsthal Lucas numbers at positive and negative indices. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-modified Pell numbers with k-balancing and k-Lucas-balancing numbers.

scholarly journals Gaussian Generalized Tetranacci Numbers

2019 ◽  
pp. 1-21 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Lucas Numbers ◽  
Matrix Formulation ◽  
Summation Formulas ◽  
Special Cases

In this paper, we dene Gaussian generalized Tetranacci numbers and as special cases, we investigate Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers with their properties. We present Binet's formulas, generating functions, and the summation formulas for Gaussian generalized Tetranacci numbers.Moreover, we give some identities connecting Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers. Furthermore, we present matrix formulation of Gaussian generalized Tetranacci numbers.

scholarly journals GENERATING FUNCTIONS OF EVEN AND ODD GAUSSIAN NUMBERS AND POLYNOMIALS

2021 ◽  
Vol 21 (1) ◽  
pp. 125-144
Author(s):  
NABIHA SABA ◽  
ALI BOUSSAYOUD ◽  
MOHAMED KERADA
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
New Class ◽  
Gaussian Polynomials

In this study, we introduce a new class of generating functions of odd and even Gaussian (p,q)-Fibonacci numbers, Gaussian (p,q)-Lucas numbers, Gaussian (p,q)-Pell numbers, Gaussian (p,q)-Pell Lucas numbers, Gaussian Jacobsthal numbers and Gaussian Jacobsthal Lucas numbers and we will recover the new generating functions of some Gaussian polynomials at odd and even terms. The technique used her is based on the theory of the so called symmetric functions.

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