The Boubaker polynomials ($B_n (X)$ are a polynomial sequence with integer coefficients. They were established in an applied physics study. The monomial expression of the Boubaker polynomials is :

$$B_n(X)=\sum_{p = 0}^{\xi(n)} (-1)^p \frac{(n-4p)}{(n-p)}\binom{n-p}{p}(X)^{n-2p}$$

where : $$\xi(n)=\left \lfloor \frac{n}{2} \right \rfloor =\frac{2n+((-1)^n - 1)}{4}$$ (The symbol :$\lfloor . \rfloor$ designates the floor function.)

The ordinary generating function of the Boubaker polynomials is : $$f_B(X,t)=\frac{1+3t^2}{1+t(t-X)}$$ The first few Boubaker polynomials are: $$B_0(X)=1$$ $$B_1(X)=X$$ $$B_2(X)=X^2+2$$ $$B_3(X)=X^3+X$$ $$B_4(X)=X^4-2$$ $$B_5(X)=X^5-X^3-3X$$ $$B_6(X)=X^6-2X^4-3X^2+2$$ $$B_7(X)=X^7-3X^5-2X^3+5X$$ $$B_8(X)=X^8-4X^6+8X^2-2$$ $$B_9(X)=X^9-5X^7+3X^5+10X^3-7X$$ $$...$$ $$...$$

Boubaker polynomials have their characteristic homogenous differential equation.

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

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

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Labiadh H. et al.
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Boubaker K.
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The OnLine Encyclopedia of Integer Sequences.
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