# Chebyshev polynomial

The *Chebyshev polynomials of first kind ^{}* are defined by the simple
formula

$${T}_{n}(x)=\mathrm{cos}(nt),$$ |

where $x=\mathrm{cos}t$.

It is an example of a *trigonometric polynomial*.

This can be seen to be a polynomial^{} by expressing $\mathrm{cos}(kt)$ as a polynomial of $\mathrm{cos}(t)$, by using the formula for cosine of angle-sum:

$\mathrm{cos}(1t)$ | $=$ | $\mathrm{cos}(t)$ | ||

$\mathrm{cos}(2t)$ | $=$ | $\mathrm{cos}(t)\mathrm{cos}(t)-\mathrm{sin}(t)\mathrm{sin}(t)=2{(\mathrm{cos}(t))}^{2}-1$ | ||

$\mathrm{cos}(3t)$ | $=$ | $4{(\mathrm{cos}(t))}^{3}-3\mathrm{cos}(t)$ | ||

$\mathrm{\vdots}$ |

So we have

${T}_{0}(x)$ | $=$ | $1$ | ||

${T}_{1}(x)$ | $=$ | $x$ | ||

${T}_{2}(x)$ | $=$ | $2{x}^{2}-1$ | ||

${T}_{3}(x)$ | $=$ | $4{x}^{3}-3x$ | ||

$\mathrm{\vdots}$ |

These polynomials obey the recurrence relation:

$${T}_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{T}_{n}(x)-{T}_{n-1}(x)$$ |

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots}$

Related are the *Chebyshev polynomials of the second kind* that are
defined as

$${U}_{n-1}(\mathrm{cos}t)=\frac{\mathrm{sin}(nt)}{\mathrm{sin}(t)},$$ |

which can similarly be seen to be polynomials through either a similar process as the above or by the relation ${U}_{n-1}(t)=n{T}_{n}^{\prime}(t)$.

The first few are:

${U}_{0}(x)$ | $=$ | $1$ | ||

${U}_{1}(x)$ | $=$ | $2x$ | ||

${U}_{2}(x)$ | $=$ | $4{x}^{2}-1$ | ||

${U}_{3}(x)$ | $=$ | $8{x}^{3}-4x$ | ||

$\mathrm{\vdots}$ |

The same recurrence relation also holds for $U$:

$${U}_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{U}_{n}(x)-{U}_{n-1}(x)$$ |

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots}$.

Title | Chebyshev polynomial |
---|---|

Canonical name | ChebyshevPolynomial |

Date of creation | 2013-03-22 12:22:56 |

Last modified on | 2013-03-22 12:22:56 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 11 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 42C05 |

Classification | msc 42A05 |

Classification | msc 33C45 |

Related topic | Polynomial |

Defines | Chebyshev polynomial of first kind |

Defines | Chebyshev polynomial of second kind^{} |