# Definition:Boubaker Polynomials

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## Definition

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

\(\ds \map {B_0} x\) | \(=\) | \(\ds 1\) | ||||||||||||

\(\ds \map {B_1} x\) | \(=\) | \(\ds x\) | ||||||||||||

\(\ds \map {B_2} x\) | \(=\) | \(\ds x^2 + 2\) | ||||||||||||

\(\ds \map {B_3} x\) | \(=\) | \(\ds x^3 + x\) | ||||||||||||

\(\ds \map {B_4} x\) | \(=\) | \(\ds x^4 - 2\) | ||||||||||||

\(\ds \map {B_5} x\) | \(=\) | \(\ds x^5 - x^3 - 3 x\) | ||||||||||||

\(\ds \map {B_6} x\) | \(=\) | \(\ds x^6 - 2 x^4 - 3 x^2 + 2\) | ||||||||||||

\(\ds \map {B_7} x\) | \(=\) | \(\ds x^7 - 3 x^5 - 2 x^3 + 5 x\) | ||||||||||||

\(\ds \map {B_8} x\) | \(=\) | \(\ds x^8 - 4 x^6 + 8 x^2 - 2\) | ||||||||||||

\(\ds \map {B_9} x\) | \(=\) | \(\ds x^9 - 5 x^7 + 3 x^5 + 10 x^3 - 7 x\) | ||||||||||||

\(\ds \) | \(\vdots\) | \(\ds \) |

## Recursive Definition

The **Boubaker polynomials ** are defined as:

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

## Closed Form

The **Boubaker polynomials ** are defined in closed form as:

- $\ds \map {B_n} x = \sum_{p \mathop = 0}^{\floor {n / 2} } \frac {n - 4 p} {n - p} \binom {n - p} p \paren {-1}^p x^{n - 2 p}$

## From Differential Equation

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

- $\paren {x^2 - 1} \paren {3 n x^2 + n - 2} \dfrac {\rd^2 y} {\rd x^2} + 3 x \paren {n x^2 + 3 n - 2} \dfrac {\rd y} {\rd x} - n \paren {3 n^2 x^2 + n^2 - 6 n + 8} y = 0$

## Also see

- Equivalence of Definitions of Boubaker Polynomials
- Results about
**Boubaker Polynomials**can be found here.

## Source of Name

This entry was named for Boubaker Boubaker.