scholarly journals On Binomial Transform of the Generalized Fifth Order Pell Sequence

2021 ◽  
pp. 8-29
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Lucas Sequences ◽  
Summation Formulas ◽  
Special Cases ◽  
Binomial Transform ◽  
Fifth Order

In this paper, we define the binomial transform of the generalized fifth order Pell sequence and as special cases, the binomial transform of the fifth order Pell and fifth order Pell-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’s formulas, generating functions, Simson formulas, recurrence properties, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.

scholarly journals Binomial Transform of the Generalized Tribonacci Sequence

2020 ◽  
pp. 26-55
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Generating Functions ◽  
Lucas Sequences ◽  
Summation Formulas ◽  
Special Cases ◽  
Binomial Transform

In this paper, we define the binomial transform of the generalized Tribonacci sequence and as special cases, the binomial transform of the Tribonacci, Tribonacci-Lucas, Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.

scholarly journals Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences

2019 ◽  
pp. 1-14
Author(s):  
Yüksel Soykan
Keyword(s):  
Linear Recurrence ◽  
Order Linear ◽  
Lucas Sequences ◽  
Summation Formulas ◽  
Special Cases ◽  
Recurrence Sequences ◽  
Fifth Order

In this paper, closed forms of the summation formulas for generalized Pentanacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Pentanacci, Pentanacci-Lucas, fifth order Pell, fifth order Pell-Lucas, fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences. We present the proofs to indicate how these formulas, in general, were discovered. In fact, all the listed formulas of the special cases of of the main theorems may be proved by induction, but that method of proof gives no clue about their discovery.

scholarly journals On Generalized p-Mersenne Numbers

2021 ◽  
pp. 83-120
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Lucas Sequences ◽  
Mersenne Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized p-Mersenne sequence and deal with, in detail, two special cases, namely, p-Mersenne and p-Mersenne-Lucas-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 On Generalized Grahaml Numbers

2020 ◽  
pp. 42-57
Author(s):  
Yuksel Soykan
Keyword(s):  
Generating Functions ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized Grahaml sequences and we deal with, in detail, three special cases which we call them Grahaml, Grahaml-Lucas and modified Grahaml 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 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 Properties of Generalized 6-primes Numbers

2020 ◽  
pp. 12-30
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized 6-primes sequence and we deal with, in detail, three special cases which we call them 6-primes, Lucas 6-primes and modified 6-primes 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 A STUDY ON GENERALIZED 5-PRIMES NUMBERS

2020 ◽  
Vol 4 (3) ◽  
pp. 185-202
Author(s):  
Yüksel SOYKAN
Keyword(s):  
Generating Functions ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized 5-primes numbers sequences and we deal with, in detail, three special cases which we call them 5-primes, Lucas 5-primes and modified 5-primes 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 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 A Study on Generalized p-Oresme Numbers

2021 ◽  
pp. 1-25
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized p-Oresme sequences and we deal with, in detail, three special cases which we call them modified p-Oresme, p-Oresme-Lucas and p-Oresme 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.

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