scholarly journals Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: The Sum Formula ∑nk=0 xkW2mk+j

2021 ◽  
pp. 30-62
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Arithmetic Progression ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases ◽  
Sum Formula

In this paper, closed forms of the sum formulas ∑n k=0 xkWmk 2 +j for generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.

scholarly journals On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0 kW3 k and Σn k=1 kW3− k

2020 ◽  
pp. 37-52 ◽  
Author(s):  
Yuksel Soykan
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas Σn k=0 kW3 k and Σn k=1 kW3-k for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

scholarly journals Corrigendum: On Summing Formulas for Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers

2020 ◽  
pp. 66-82
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers and Gaussian Fibonacci, Gaussian Lucas, Gaussian Pell, Gaussian Pell-Lucas, Gaussian Jacobsthal, Gaussian Jacobsthal-Lucas numbers.

scholarly journals Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0W3 Σ k and n k=1W3

2020 ◽  
pp. 58-69
Author(s):  
Y¨ uksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the sum formulas for the cubes of generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas numbers.

scholarly journals On Summing Formulas for Horadam Numbers

2020 ◽  
pp. 45-61
Author(s):  
Y¨ uksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas for generalized Fibonacci numbers arepresented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas,Jacobsthal, Jacobsthal-Lucas numbers.

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 Generalized Fibonacci Numbers: Sum Formulas

2020 ◽  
pp. 89-104
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.

scholarly journals A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $

2020 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3}$ for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.

scholarly journals A Study on the Sums of Squares of Generalized Fibonacci Numbers: Closed Forms of the Sum Formulas Σn k=0 kxkW2 k and Σn k=1 kxkW2- k

2020 ◽  
pp. 44-67
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas Σn k=0 kxkW2 k and Σn k=1 kxkW2 -k for the squares of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

scholarly journals Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j

2021 ◽  
pp. 11-38
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.

scholarly journals On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers

2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers and Gaussian Fibonacci, Gaussian Lucas, Gaussian Pell, Gaussian Pell-Lucas, Gaussian Jacobsthal, Gaussian Jacobsthal-Lucas numbers.

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