scholarly journals The formulas for sum of products of sequences associated with the metallic means

2020 ◽  
Vol 1 (1) ◽  
pp. 73-78
Author(s):  
P. Kosobutskyy ◽  
N. Nestor
Keyword(s):  
Signal Processing ◽  
Closed Form ◽  
Fibonacci Numbers ◽  
Statistical Signal Processing ◽  
Discrete Data ◽  
Definite Integrals ◽  
Sum Of Products ◽  
Finite Sums ◽  
Discrete Analogs

In this paper, the regularities of convolution of sequences c of Fibonacci numbers {Fn} generated by metallic means and the sum of products of two statistically independent sequences {Fi} and Jn=j∙sin(0.5π(n-j)) are investigated. It is shown that the known closed forms of sums for convolution and product are similar. Attention to the study of the convolution of two sequences of discrete data is associated with the use of this method for statistical signal processing. This problem involves calculating finite sums as discrete analogs of definite integrals. Such a problem is considered solved if the formula for the sum is expressed in a closed form as a function of its members and their number.

scholarly journals Simultaneous Contour Method of a Trigonometric Integral by Prudnikov et al.

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1453
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Contour Method ◽  
Substantial Part ◽  
Exponential Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Transcendental Functions ◽  
Trigonometric Integral

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

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.

FAULT DIAGNOSIS OF MULTITOOTH MACHINE TOOL BASED ON STATISTICAL SIGNAL PROCESSING

2002 ◽  
Vol 35 (1) ◽  
pp. 197-202 ◽  
Author(s):  
Aníbal Reñones Domínguez ◽  
Luis J. De Miguel González
Keyword(s):  
Signal Processing ◽  
Fault Diagnosis ◽  
Machine Tool ◽  
Statistical Signal Processing
Sign in / Sign up

Export Citation Format

Share Document