18 Orthogonal PolynomialsClassical Orthogonal Polynomials18.7 Interrelations and Limit Relations18.9 Recurrence Relations and Derivatives

$f\left(x\right)$ | $A\left(x\right)$ | $B\left(x\right)$ | $C\left(x\right)$ | ${\lambda}_{n}$ |
---|---|---|---|---|

${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$ | $1-{x}^{2}$ | $\beta -\alpha -\left(\alpha +\beta +2\right)x$ | 0 | $n\left(n+\alpha +\beta +1\right)$ |

${\left(\mathrm{sin}\frac{1}{2}x\right)}^{\alpha +\frac{1}{2}}{\left(\mathrm{cos}\frac{1}{2}x\right)}^{\beta +\frac{1}{2}}\times {P}_{n}^{\left(\alpha ,\beta \right)}\left(\mathrm{cos}x\right)$ | $1$ | $0$ | $\frac{\frac{1}{4}-{\alpha}^{2}}{4{\mathrm{sin}}^{2}\frac{1}{2}x}}+{\displaystyle \frac{\frac{1}{4}-{\beta}^{2}}{4{\mathrm{cos}}^{2}\frac{1}{2}x}$ | ${\left(n+\frac{1}{2}\left(\alpha +\beta +1\right)\right)}^{2}$ |

${\left(\mathrm{sin}x\right)}^{\alpha +\frac{1}{2}}{P}_{n}^{\left(\alpha ,\alpha \right)}\left(\mathrm{cos}x\right)$ | $1$ | $0$ | $\left(\frac{1}{4}-{\alpha}^{2}\right)/{\mathrm{sin}}^{2}x$ | ${\left(n+\alpha +\frac{1}{2}\right)}^{2}$ |

${C}_{n}^{\left(\lambda \right)}\left(x\right)$ | $1-{x}^{2}$ | $-\left(2\lambda +1\right)x$ | $0$ | $n\left(n+2\lambda \right)$ |

${T}_{n}\left(x\right)$ | $1-{x}^{2}$ | $-x$ | $0$ | ${n}^{2}$ |

${U}_{n}\left(x\right)$ | $1-{x}^{2}$ | $-3x$ | $0$ | $n\left(n+2\right)$ |

${P}_{n}\left(x\right)$ | $1-{x}^{2}$ | $-2x$ | $0$ | $n\left(n+1\right)$ |

${L}_{n}^{\left(\alpha \right)}\left(x\right)$ | $x$ | $\alpha +1-x$ | $0$ | $n$ |

${\mathrm{e}}^{-\frac{1}{2}{x}^{2}}{x}^{\alpha +\frac{1}{2}}{L}_{n}^{\left(\alpha \right)}\left({x}^{2}\right)$ | $1$ | $0$ | $-{x}^{2}+\left(\frac{1}{4}-{\alpha}^{2}\right){x}^{-2}$ | $4n+2\alpha +2$ |

${H}_{n}\left(x\right)$ | $1$ | $-2x$ | $0$ | $2n$ |

${\mathrm{e}}^{-\frac{1}{2}{x}^{2}}{H}_{n}\left(x\right)$ | $1$ | $0$ | $-{x}^{2}$ | $2n+1$ |

${\mathit{He}}_{n}\left(x\right)$ | $1$ | $-x$ | $0$ | $n$ |