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Equation Expression Differential Equation Formal Power Series Characteristic Floor Boubaker Turki Polynomials Boubaker Polynomials

1	Expression
1	Equation
2	Differential Equation
2	Formal Power Series
3	Characteristic
4	Floor
14	Boubaker Polynomials
15	Boubaker Turki Polynomials

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Boubaker Turki Polynomials

The Boubaker-Tuki polynomials (or Modified Boubaker polynomials ,noted: or $\tilde {B}_n (X)$ ) are an enhanced form of the Boubaker polynomials. They were named after Pr. Boubaker-Mustapha Rachid TURKI (1934-1987). The Boubaker-Turki polynomials are solutions of the characterisric Sturm-Liouville differential equation:

$\displaystyle (X^2-1)(3nX^2+n-2)y{''}+3X(nX^2+3n-2)y{'}-n(3n^2X^2+n^2-6n+8)y=0$

(Being the unique solution to this equation hasn't been demonstrated)

The explicit expression of the Boubaker-Tuki polynomials is:

$\displaystyle \tilde {B}_n(X)=2^{n}X^{n}-2^{n-2}(n-4)X^{n-2}+\sum_{p = 2}^{\xi(n)} (-1)^p \frac{n-4p}{p!}\prod_{j=p+1}^{2p-1}(n-j).2^{n-2p}X^{n-2p}$

where :

$\displaystyle \xi(n)=\left \lfloor \frac{n}{2} \right \rfloor =\frac{2*n+((-1)^n-1)}{4}$

(The symbol : $\lfloor \rfloor$ designates the floor function.)

The Boubaker-Tuki polynomials have their characteristic ordinary generating function.

Bibliography

1

Labiadh H. et al.
A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials
Journal of Differential Equations and Control Processes.,37: 105/109.2007.
Available online

2

Bamidele Awojoyogbe O (Nigeria).
A solution to Bloch NMR flow equations for the analysis of homodynamic functions of blood flow system using m-Boubaker polynomials
Current Applied Physics (Elsevier),DOI: 10.1016/j.cap.2008.01.019.
Available online

3

Labiadh H. et al.
Establishment of an Ordinary Generating Function and a Christoffel-Darboux Type First-Order Differential Equation for the Heat Equation Related Boubaker-Tuki polynomials
Journal of Differential Equations and Control Processes.,1, 51/66. 2008.
Available online

4

Karem B.
On Modified Boubaker polynomials : Some Differential and Analytical Properties of the New Polynomials Issued from an Attempt for Solving Bi-varied Heat Equation.
Trends in Applied Sciences Research.,2 (6): 540/544.
Available online

5

The OnLine Encyclopedia of Integer Sequences.
Sequence number: A135929Link,A135936Link,A137276Link ,A137277Link ,A137289Link ,A138034Link. OLEIS,A138034,Sequences related to Boubaker polynomials Bn(X) and Boubaker-Turki polynomials mBn(X) (by Roger L. Bagula, A. bannour,G. Adamson and njas).
Link

6

The OnLine Encyclopedia of Integer Sequences.
OLEIS,A138476,Sequences related to the Boubaker-Turki polynomials mBn(X).
Sequence number: A138476
Link