Algebraic Independence of Sums of Reciprocals of the Fibonacci Numbers

1999 ◽  
Vol 202 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Kumlko Nishioka ◽  
Taka-Aki Tanaka ◽  
Takeshi Toshimitsu
Keyword(s):  
Fibonacci Numbers ◽  
Algebraic Independence

scholarly journals Algebraic independence of infinite products generated by Fibonacci numbers

2011 ◽  
Vol 34 (2) ◽  
pp. 255-264 ◽  
Author(s):  
Takeshi Kurosawa ◽  
Yohei Tachiya ◽  
Taka-aki Tanaka
Keyword(s):  
Fibonacci Numbers ◽  
Algebraic Independence ◽  
Infinite Products

Algebraic independence results for reciprocal sums of Fibonacci numbers

2011 ◽  
Vol 148 (3) ◽  
pp. 205-223 ◽  
Author(s):  
Carsten Elsner ◽  
Shun Shimomura ◽  
Iekata Shiokawa
Keyword(s):  
Fibonacci Numbers ◽  
Algebraic Independence ◽  
Independence Results ◽  
Reciprocal Sums

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Algebraic Independence Results Related to 〈q, r〉-Number Systems

2006 ◽  
Vol 147 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Shin-ichiro Okada ◽  
Iekata Shiokawa
Keyword(s):  
Algebraic Independence ◽  
Number Systems ◽  
Independence Results
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