Q_n^(α, β)(x) = 2^(-n - 1) (x - 1)^(-α) (x + 1)^(-β) × integral_(-1)^1 (1 - t)^(n + α) (1 + t)^(n + β) (x - t)^(-n - 1) d t. In the exceptional case n = 0, α + β + 1 = 0, a nonconstant solution is given by Q^(α)(x) = ln(x + 1) + π^(-1) sin(πα)(x - 1)^(-α) (x + 1)^(-β) × integral_(-1)^1 ((1 - t)^α (1 + t)^β)/(x - t) ln(1 + t) d t.

Jacobi differential equation | Jacobi polynomial