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Table 1 Leading Terms for Classical Polynomials

From: Polynomial solutions of differential equations

Polynomial a n b n c n
Legendre 1 0 n 2 ( 2 n + 1 ) ( 2 n - 1 )
Hermite 1 0 n 2
Laguerre 1 2n + 1 n 2
Chebychev 1 0 1 4 ( for n 2 )
Jacobi 1 - β ( 2 + α ) ( 2 n - 2 - α ) ( 2 n - α ) n ( n - α - 2 ) ( 2 n - ( β + α + 2 ) ) ( 2 n + ( β - α - 2 ) ) ( 2 n - α - 3 ) ( 2 n - 2 - α ) 2 ( 2 n - α - 1 ) (for n ≥ 2)
   b 1 = β ( 2 + α ) α ( 2 - α ) c 1 = ( α - β ) ( α + β ) ( 1 - α ) α 2
    : here α <β < - α