Also known as the Hypergeometric Polynomials, they occur in the study of
Rotation Groups and in the solution to the equations of motion of the symmetric
top. They are solutions to the Jacobi Differential Equation. Plugging

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

Jacobi polynomials are Orthogonal satisfying

(9) |

(10) |

(11) |

(12) |

The Derivative is given by

(13) |

The Orthogonal Polynomials with Weighting Function
on the
Closed Interval can be expressed in the form

(14) |

Special cases with are

(15) | |||

(16) | |||

(17) | |||

(18) |

Further identities are

(19) |

(20) |

(21) |

The Kernel Polynomial is

(22) |

The Discriminant is

(23) |

For
,
reduces to a Legendre Polynomial. The Gegenbauer Polynomial

(24) |

(25) |

(26) |

Let be the number of zeros in , the number of zeros in
, and the
number of zeros in
. Define Klein's symbol

(27) |

(28) | |||

(29) | |||

(30) |

If the cases , , ..., , , , ..., , and , , ..., are excluded, then the number of zeros of in the respective intervals are

(31) | |||

(32) | |||

(33) |

(Szegö 1975, pp. 144-146).

The first few Polynomials are

(34) | |||

(35) | |||

(36) |

where is a Rising Factorial. See Abramowitz and Stegun (1972, pp. 782-793) and Szegö (1975, Ch. 4) for additional identities.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 771-802, 1972.

Iyanaga, S. and Kawada, Y. (Eds.). ``Jacobi Polynomials.'' Appendix A, Table 20.V in
*Encyclopedic Dictionary of Mathematics.* Cambridge, MA: MIT Press, p. 1480, 1980.

Szegö, G. ``Jacobi Polynomials.'' Ch. 4 in *Orthogonal Polynomials, 4th ed.* Providence, RI: Amer. Math. Soc., 1975.

© 1996-9

1999-05-25