 The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of formula_26 of the generating function

The coefficient of formula_27 is a polynomial in formula_28 of degree formula_4.Expanding up to formula_30 gives
Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.It is possible to obtain the higher formula_14's without resorting to direct expansion of the Taylor series, however.