JacobiP - Maple Programming Help

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JacobiP

Jacobi function

 Calling Sequence JacobiP(n, a, b, x)

Parameters

 n - algebraic expression a - nonrational algebraic expression or rational number greater than -1 b - nonrational algebraic expression or rational number greater than -1 x - algebraic expression

Description

 • 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 $\left[-1,1\right]$ 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:

${\int }_{-1}^{1}w\left(t\right)\mathrm{JacobiP}\left(m,a,b,t\right)\mathrm{JacobiP}\left(n,a,b,t\right)ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{{2}^{a+b+1}\mathrm{\Gamma }\left(n+a+1\right)\mathrm{\Gamma }\left(n+b+1\right)}{\left(2n+a+b+1\right)n!\mathrm{\Gamma }\left(n+a+b+1\right)}& n=m\end{array}$

 The Jacobi polynomials are undefined for negative integer values of a or b.
 • The polynomials satisfy the following recurrence relation:

$\mathrm{JacobiP}\left(0,a,b,x\right)=1$

$\mathrm{JacobiP}\left(1,a,b,x\right)=\frac{1}{2}a-\frac{1}{2}b+\left(1+\frac{1}{2}a+\frac{1}{2}b\right)x$

$\mathrm{JacobiP}\left(n,a,b,x\right)=\frac{1}{2}\frac{\left(2n+a+b-1\right)\left({a}^{2}-{b}^{2}+\left(2n+a+b-2\right)\left(2n+a+b\right)x\right)\mathrm{JacobiP}\left(n-1,a,b,x\right)}{n\left(n+a+b\right)\left(2n+a+b-2\right)}-\frac{\left(n+a-1\right)\left(n+b-1\right)\left(2n+a+b\right)\mathrm{JacobiP}\left(n-2,a,b,x\right)}{n\left(n+a+b\right)\left(2n+a+b-2\right)},\mathrm{for n > 1.}$

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

$\mathrm{JacobiP}\left(n,a,b,x\right)=\mathrm{binomial}\left(a+n,a\right)\mathrm{hypergeom}\left(\left[-n,a+b+n+1\right],\left[a+1\right],\frac{1}{2}-\frac{1}{2}x\right)$

Examples

 > $\mathrm{JacobiP}\left(4,1,\frac{3}{4},x\right)$
 ${\mathrm{JacobiP}}{}\left({4}{,}{1}{,}\frac{{3}}{{4}}{,}{x}\right)$ (1)
 > $\mathrm{simplify}\left(,'\mathrm{JacobiP}'\right)$
 ${-}\frac{{115}}{{4}}{+}\frac{{135}}{{4}}{}{x}{+}\frac{{4185}}{{64}}{}{\left({x}{-}{1}\right)}^{{2}}{+}\frac{{48825}}{{1024}}{}{\left({x}{-}{1}\right)}^{{3}}{+}\frac{{380835}}{{32768}}{}{\left({x}{-}{1}\right)}^{{4}}$ (2)
 > $\mathrm{JacobiP}\left(2.2,1,\frac{2}{3},0.4\right)$
 ${-}{0.1993478307}$ (3)