scholarly journals Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

2017 ◽  
Vol 57 (1) ◽  
pp. 1-84
Author(s):  
Alexandre Laugier ◽  
Manjil P. Saikia
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Polynomial Sequences ◽  
Fibonacci Polynomial

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

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 Aitken sequences and generalized Fibonacci numbers

1985 ◽  
Vol 45 (172) ◽  
pp. 553-553 ◽  
Author(s):  
J. H. McCabe ◽  
G. M. Phillips
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers

scholarly journals On the Sum of Reciprocal Generalized Fibonacci Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Pingzhi Yuan ◽  
Zilong He ◽  
Junyi Zhou
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Infinite Sums

We consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.

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