Re: Representation of Jacobi polynomials

["Daniel Lozier" <lozier@nist.gov>]

Sender: Victor Adamchik <adamchik@ux10.sp.cs.cmu.edu>
Subject: Re: Representation of Jacobi polynomials


For a=b, the Jacobi function can be expressed in terms of
the Gegenbauer C-function. Here is the formula

P_{n}^{(a,a)}(z) = \frac{(a+1)_{n}}{(2a+1)_{n}} * C_{n}^{a+1/2}(z)

where (a+1)_{n} is the Pochhammer symbol, and n is any.
A few particular cases
  a=1/2,    P_{n}^{(a,a)}(z) ~ Chebyshev U
  a=-1/2,   P_{n}^{(a,a)}(z) ~ Chebyshev T

For a=-b, the Jacobi function is reduced to the Legendre P function.

For arbitrary values of parameters 'a' and 'b', but integer and positive
n, the Jacobi function is a finite sum

P_{n}^{(a,b)}(z) = \frac{1}{2^n} \sum_{k=0}^{k=n} \choose{a+n,k}
\choose{b+n,n-k} *(z+1)^{k} * (z-1)^{n-k}

Victor S. Adamchik