Table 1 Leading Terms for Classical Polynomials

Legendre

1

0

$\frac{n^2}{\left(2n+1\right)\left(2n-1\right)}$

Hermite

1

0

$\frac{n}{2}$

Laguerre

1

2n + 1

n ²

Chebychev

1

0

$\frac{1}{4}\left(\mathsf{\text{for}}n\ge 2\right)$

Jacobi

1

$\frac{-\beta \left(2+\alpha \right)}{\left(2n-2-\alpha \right)\left(2n-\alpha \right)}$

$\frac{n\left(n-\alpha -2\right)\left(2n-\left(\beta +\alpha +2\right)\right)\left(2n+\left(\beta -\alpha -2\right)\right)}{\left(2n-\alpha -3\right){\left(2n-2-\alpha \right)}^2\left(2n-\alpha -1\right)}$ (for n ≥ 2)

$b_1=\frac{\beta \left(2+\alpha \right)}{\alpha \left(2-\alpha \right)}$

$c_1=\frac{\left(\alpha -\beta \right)\left(\alpha +\beta \right)}{\left(1-\alpha \right){\alpha }^2}$

: here α <β < - α