The P(n, a, b, x) function computes the nth Jacobi polynomial with parameters a and b evaluated at x.

In the case of only two arguments, P(n, x) computes the nth Legendre (spherical) polynomial which is equal to P(n, 0, 0, x).

These polynomials are orthogonal on the interval $\left[\mathrm{-1}\,1\right]$ with respect to the weight function $w\left(x\right)={\left(1-x\right)}^a{\left(1+x\right)}^b$ when a and b are greater than -1. They satisfy:

The Jacobi polynomials are undefined for negative integer values of a or b.

Jacobi polynomials satisfy the following recurrence relation:

$\mathrm{with}\left(\mathrm{orthopoly}\right)\:$

$P\left(3\,x\right)$

$\frac{5}{2}{x}^3-\frac{3}{2}x$

$P\left(30\,\frac{1}{3}\right)$

$\frac{18024734042221}{205891132094649}$

$P\left(4\,1\,\frac{3}{4}\,x\right)$

$-\frac{115}{4}+\frac{135x}{4}+\frac{4185{\left(x-1\right)}^2}{64}+\frac{48825{\left(x-1\right)}^3}{1024}+\frac{380835{\left(x-1\right)}^4}{32768}$

$P\left(7\,-\frac{2}{3}\,\frac{7}{4}\,\frac{1}{2}\right)$

$-\frac{7258990337}{38654705664}$