scholarly journals A Study on Sum Formulas of Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas \(\sum_{k=0}^{n}kW_{k}\) and \(\sum_{k=1}^{n}kW_{-k}\)

2021 ◽  
pp. 68-85
Author(s):  
Y¨ uksel Soykan
Keyword(s):  
Fourth Order ◽  
Summation Formulas ◽  
Special Cases ◽  
Recurrence Sequences

In this paper, closed forms of the sum formulas \(\sum_{k=0}^{n}kW_{k}\) and \(\sum_{k=1}^{n}kW_{-k}\) for generalized Tetranacci numbers are presented. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, and other fourth-order recurrence sequences.

scholarly journals A Study on Sum Formulas of Generalized Hexanacci Numbers: Closed Forms of the Sum Formulas ∑n k=0 xkWk and ∑nk=1 xkW−k

2021 ◽  
pp. 93-118
Author(s):  
Yüksel Soykan
Keyword(s):  
Sixth Order ◽  
Summation Formulas ◽  
Special Cases ◽  
Recurrence Sequences

In this paper, closed forms of the sum formulas ∑n k =0 xkWk and ∑n k=1 xkW-k for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, Hexanacci-Lucas, and other sixth-order recurrence sequences.

scholarly journals A Study On Sum Formulas of Generalized Sixth-Order Linear Recurrence Sequences

2020 ◽  
pp. 36-48
Author(s):  
Yüksel Soykan
Keyword(s):  
Sixth Order ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Order Linear ◽  
Summation Formulas ◽  
Special Cases ◽  
Recurrence Sequences

In this paper, closed forms of the summation formulas for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, Hexanacci-Lucas, sixth order Pell, sixth order Pell-Lucas, sixth order Jacobsthal, sixth order Jacobsthal-Lucas numbers.

scholarly journals A Study on Generalized Tetranacci Numbers: Closed Form Formulas ∑n k=0 xkWk2 of Sums of the Squares of Terms

2020 ◽  
pp. 109-136
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Closed Form ◽  
Fourth Order ◽  
Order Linear ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the sum formulas ∑n k=0 xkWk2 for the squares of generalized Tetranacci numbers are presented. We also present the sum formulas ∑n k=0 xkWk+1Wk; ∑n k=0 xkWk+2Wk; and ∑n k=0 xkWk+3Wk: As special cases, we give summation formulas of the of Tetranacci, Tetranacci-Lucas and some other fourth order linear recurrance sequences.

scholarly journals On Generalized Tetranacci Numbers: Closed Form Formulas of the Sum $\sum_{k=0}^{n}W_{k}^{2}$ of the Squares of Terms

2020 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Closed Form ◽  
Fourth Order ◽  
Order Linear ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}W_{k}^{2}$ \ for the squares of generalized Tetranacci numbers are presented. We also present the sum formulas $\sum_{k=0}^{n}W_{k+1}W_{k},$ $\sum_{k=0}^{n}W_{k+2}W_{k},$ and $\sum_{k=0}^{n}W_{k+3}W_{k}.$ As special cases, we give summation formulas of the of Tetranacci, Tetranacci-Lucas and some other fourth order linear recurrance sequences.

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 Summation Formulas for Generalized Tetranacci Numbers

2019 ◽  
pp. 1-12
Author(s):  
Yüksel Soykan
Keyword(s):  
Fourth Order ◽  
Lucas Numbers ◽  
Summation Formulas ◽  
Special Cases

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

scholarly journals Simplified and improved criteria for oscillation of delay differential equations of fourth order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. Moaaz ◽  
A. Muhib ◽  
D. Baleanu ◽  
W. Alharbi ◽  
E. E. Mahmoud
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Interesting Point ◽  
All Solutions ◽  
Special Cases ◽  
Delay Differential ◽  
Noncanonical Operator

AbstractAn interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

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 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.

Divisor Sums of Generalised Exponential Polynomials

1996 ◽  
Vol 39 (1) ◽  
pp. 35-46 ◽  
Author(s):  
G. R. Everest ◽  
I. E. Shparlinski
Keyword(s):  
Prime Ideal ◽  
Algebraic Integer ◽  
Linear Recurrence ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Recurrence Sequence ◽  
Special Cases ◽  
Divisor Sums ◽  
Recurrence Sequences

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

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