# Chebyshev equation

Chebyshev’s equation is the second order^{} linear differential equation

$$(1-{x}^{2})\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{p}^{2}y=0$$ |

where $p$ is a real constant.

There are two independent solutions which are given as series by:

$${y}_{1}(x)=1-\frac{{p}^{2}}{2!}{x}^{2}+\frac{(p-2){p}^{2}(p+2)}{4!}{x}^{4}-\frac{(p-4)(p-2){p}^{2}(p+2)(p+4)}{6!}{x}^{6}+\mathrm{\cdots}$$ |

and

$${y}_{2}(x)=x-\frac{(p-1)(p+1)}{3!}{x}^{3}+\frac{(p-3)(p-1)(p+1)(p+3)}{5!}{x}^{5}-\mathrm{\cdots}$$ |

In each case, the coefficients are given by the recursion

$${a}_{n+2}=\frac{(n-p)(n+p)}{(n+1)(n+2)}{a}_{n}$$ |

with ${y}_{1}$ arising from the choice ${a}_{0}=1$, ${a}_{1}=0$, and ${y}_{2}$ arising from the choice ${a}_{0}=0$, ${a}_{1}=1$.

The series converge^{} for $$; this is easy to see from the ratio test^{} and the recursion formula^{} above.

When $p$ is a non-negative integer, one of these series will terminate, giving a polynomial solution. If $p\ge 0$ is even, then the series for ${y}_{1}$ terminates at ${x}^{p}$. If $p$ is odd, then the series for ${y}_{2}$ terminates at ${x}^{p}$.

These polynomials are, up to multiplication by a constant, the Chebyshev polynomials^{}. These are the only polynomial solutions of the Chebyshev equation.

(In fact, polynomial solutions are also obtained when $p$ is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of $p$ by $-p$.)

Title | Chebyshev equation |
---|---|

Canonical name | ChebyshevEquation |

Date of creation | 2013-03-22 13:10:17 |

Last modified on | 2013-03-22 13:10:17 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 34A30 |

Synonym | Chebyshev differential equation |

Related topic | HermiteEquation |