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If the first parameter is a non-negative integer, the JacobiP(n, a, b, x) function computes the nth Jacobi polynomial with parameters a and b evaluated at 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 following:

The polynomials satisfy the following recurrence relation:

For n and not equal to a non-negative integer and a not a negative integer, the analytic extension of the Jacobi polynomial is given by the following:

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

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

$\mathrm{simplify}\left(\,'\mathrm{JacobiP}'\right)$

$\frac{19075}{32768}-\frac{3915}{8192}x-\frac{129735}{16384}{x}^2+\frac{9765}{8192}{x}^3+\frac{380835}{32768}{x}^4$

$\mathrm{JacobiP}\left(2.2\,1\,\frac{2}{3}\,0.4\right)$

$\mathrm{-0.1993478307}$

The JacobiP command was updated in Maple 2020.