This discussion is about zero calculation Jacobi orthogonal polynomials, which can search Google Scholar relevant to many scholarly articles. Due to the recent frequent use Jacobi orthogonal polynomials Galerkin-Spectral method, so we put together a number of related skills.
Jacobi orthogonal polynomials recurrence formula:
$J_0^{\alpha,\beta}(x)=1$, $\quad J_1^{\alpha,\beta}(x)=\frac12 (\alpha+\beta+2)x+\frac12(\alpha-\beta),$
$J_{n+1}^{\alpha,\beta}(x)=\Big(a_n^{\alpha,\beta}x-b_n^{\alpha,\beta}\Big)J_{n}^{\alpha,\beta}(x)-c_{n}^{\alpha,\beta}J_{n-1}^{\alpha,\beta}(x),\quad n\geq 1.$
among them
$a_n^{\alpha,\beta}=\frac{\big( 2n+\alpha+\beta+1 \big)\big( 2n+\alpha+\beta+2 \big)}{2\big( n+1 \big)\big( n+\alpha+\beta+1 \big)},$
$b_n^{\alpha,\beta}=\frac{\big( \beta^2-\alpha^2 \big)\big( 2n+\alpha+\beta+1 \big)}{2\big( n+1 \big)\big( n+\alpha+\beta+1 \big)\big( 2n+\alpha+\beta\big)},$
$c_n^{\alpha,\beta}=\frac{\big( n+\alpha \big)\big( n+\beta\big)\big( 2n+\alpha+\beta+2 \big)}{\big( n+1 \big)\big( n+\alpha+\beta+1 \big)\big( 2n+\alpha+\beta\big)},$