Legendre polynomial of the second kind

The second solution Q_l(x) to the Legendre differential equation. The Legendre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials. The Legendre functions of the second kind are implemented in the Wolfram Language as LegendreQ[l, x]. The first few are Q_0(x) | = | 1/2 ln((1 + x)/(1 - x)) Q_1(x) | = | x/2 ln((1 + x)/(1 - x)) - 1 Q_2(x) | = | (3x^2 - 1)/4 ln((1 + x)/(1 - x)) - (3x)/2 Q_3(x) | = | (5x^3 - 3x)/4 ln((1 + x)/(1 - x)) - (5x^2)/2 + 2/3.

Legendre differential equation | Legendre function of the first kind | Legendre polynomial

LegendreQ

Adrien-Marie Legendre