Definition:Boubaker Polynomials

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Definition

The Boubaker polynomials are the components of the following sequence of polynomials:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_0 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle 1\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_1 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_2 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^2 + 2\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_3 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^3 + x\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_4 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^4 - 2\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_5 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^5 - x^3 - 3x\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_6 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^6 - 2x^4 - 3x^2 + 2\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_7 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^7 - 3x^5 - 2x^3 + 5x\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_8 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^8 - 4x^6 + 8x^2 - 2\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle B_9 \left({x}\right)\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle x^9 - 5x^7 + 3x^5 + 10x^3 - 7x\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\vdots\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)                    


Recursive Definition

The Boubaker polynomials are defined as:

$B_n \left({x}\right) = \begin{cases} 1 & : n = 0 \\ x & : n = 1 \\ x^2+2 & : n = 2 \\ x B_{n-1} \left({x}\right) - B_{n-2} \left({x}\right) & : n > 2 \end{cases}$


Closed Form

The Boubaker polynomials are defined in closed form as:

$\displaystyle B_n \left({x}\right) = \sum_{p=0}^{\lfloor n/2\rfloor} \frac {n-4p} {n-p} \binom {n-p} p \left({-1}\right)^p x^{n-2p}$


From Differential Equation

The Boubaker polynomials are defined as solutions to the differential equation:

$\displaystyle \left({x^2-1}\right) \left({3nx^2+n-2}\right) \frac {d^2y} {dx^2} + 3x \left({n x^2 + 3n - 2}\right) \frac {dy}{dx} - n \left({3n^2 x^2 + n^2 - 6n+8}\right) y = 0$


Also see

  • Results about Boubaker Polynomials can be found here.


Source of Name

This entry was named for Boubaker Boubaker.