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

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

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

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

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

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

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

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

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

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

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

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

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

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