Polinomi di Boubaker

I polinomi di Boubaker sono le componenti di una successione polinomiale che inizia con i seguenti polinomi:

${\begin{aligned}B_{0}(x)&{}=1\\B_{1}(x)&{}=x\\B_{2}(x)&{}=x^{2}+2\\B_{3}(x)&{}=x^{3}+x\\B_{4}(x)&{}=x^{4}-2\\B_{5}(x)&{}=x^{5}-x^{3}-3x\\B_{6}(x)&{}=x^{6}-2x^{4}-3x^{2}+2\\B_{7}(x)&{}=x^{7}-3x^{5}-2x^{3}+5x\\B_{8}(x)&{}=x^{8}-4x^{6}+8x^{2}-2\\B_{9}(x)&{}=x^{9}-5x^{7}+3x^{5}+10x^{3}-7x\\&{}\,\,\,\vdots \end{aligned}}$

I polinomi di Boubaker possono essere definiti in modo generalizzato tramite la formula:

${\begin{aligned}B_{0}(x)&=1,\\B_{1}(x)&=x,\\B_{2}(x)&=x^{2}+2,\\B_{m}(x)&=xB_{m-1}(x)-B_{m-2}(x)\quad {\text{for }}m>2.\end{aligned}}$

Un'altra possibile definizione di questi polinomi è la seguente:

$B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}$

Alternativamente, i polinomi di Boubaker possono essere definiti tramite l'equazione differenziale:

${\begin{aligned}(x^{2}-1)(3nx^{2}+n-2)y{''}+3x(nx^{2}+3n-2)y{'}-n(3n^{2}x^{2}+n^{2}-6n+8)y=0\,\end{aligned}}$

I polinomi di Boubaker hanno prodotto molte sequenze nell'Enciclopedia On-line di Sequenze di Numeri Interi' (OEIS) On-Line Encyclopedia of Integer Sequences e PlanetMath.

Applicazioni

I polinomi di Boubaker sono ampiamente utilizzati nei seguenti ambiti:

• criogenia
• biologia
• sistemi dinamici
• sistemi non lineari
• teoria dell'approssimazione
• termodinamica
• meccanica
• idrologia
• dinamica molecolare
• Manifoldi-->
• calorimetria
• biofisica
• fotovoltaico
• analisi complessa
• analisi di matrici
• fisica teorica
• Matematica applicata
• Crittografia
• Algebra

Bibliografia

• Encyclopedia of Physics Research

Encyclopedia of Physics Research, Chapter 21: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials, Editors: Nancy B. Devins and Jillian P. Ramos [2]

• NASA: USA-Physics Abstract Service Database

a New Parameter:. AN Abacus for Optimizing Pv-T Hybrid Solar Device Functional Materials Using the Boubaker Polynomials Expansion Scheme and citation|title= http://adsabs.harvard.edu/cgi-bin/nph-data_query?doi=10.1007/s00231-009-0493-x&db_key=PHY&link_type=ABSTRACT&high=499df58f2b32224

• La Presse

De Khawarizmi à Euler, La Presse Magazine|date=January 9, 2008 fr, vedi anche tunisie7arts.com

Le polynôme de Boubaker|journal=La Presse Magazine|date=April 22, 2007|issue=1019|page=6 [3]

• John Wiley & Sons

A new polynomial sequence... The Boubaker Polynomials|journal=Numerical Methods for Partial Differential Equations NMPDE [4]

• AIAA American Institute of Aeronautics and Astronautics, Inc.

Solution to Heat Equation Using Boubaker Polynomials|journal=J. of Thermophysics and Heat Transfer [5]

• ASME American Society of Mechanical Engineers, Inc.

Boubaker Polynomials Weak Solutions to a Robin Boundary Conditioned Dynamic-State Heat Transfer Problem |journal=International Journal of Heat Transfer [6]

• WS World Scientific Publishing Co Pte Ltd

AMLOUK–BOUBAKER EXPANSIVITY..USING BOUBAKER POLYNOMIALS ,Functional Materials Letter [7]

• Pubblicazioni accademiche

A new polynomial sequence ...The Boubaker Polynomials, International Journal of Applied Mathematics [8]

• Articoli vari in NASA Astrophysics Data System

[9][10][11][12][13][14][15][16][17][18][19][20][21][22]

• ENEA Ente Nazionale per le Energie Alternative

citation|title=An attempt... using Boubaker Polynomials|journal= International Journal of Heat and Technology [23]

• Taylor and Francis

citation|title=Numerical Distribution of Temperature During Welding Using Boubaker Polynomials , Numerical Heat Transfer, Part A, Applications [24]

• Università Statale di San Pietroburgo, Санкт-Петербургский государственный университет

Establishment of an Ordinary Generating Function and a Christoffel-Darboux Type First-Order Differential Equation for the Heat Equation Related Boubaker-Turki Polynomials, Journal of Differential Equations and C.P. [25]

Some new properties of the applied-physics related Boubaker polynomials [26]

• Slovak Academy of Sciences,Ústav stavebníctva a architektúry SAV

citation 3, Page 97 C.P., [27]

• MATH Database 1931- 2011, © European Mathematical Society, FIZ Karlsruhe & Springer-Verlag

Meixner-type results for Riordan arrays and associated integer sequences, Chap. 6 Boubaker Polynomials

An analytical solution to applied mathematics-related Love's equation using the Boubaker polynomials expansion scheme

Some new features of the Boubaker polynomials expansion scheme

The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem