Editing Legendre polynomials (section)

=== Definition via generating function ===
The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of <math>t</math> of the [[generating function]]<ref>{{harvnb|Arfken|Weber|2005|loc=p.743}}</ref>
{{NumBlk||<math display="block">\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n \,.</math>|{{EquationRef|2}}}}

The coefficient of <math>t^n</math> is a polynomial in <math> x </math> of degree <math>n</math> with <math>|x| \leq 1</math>. Expanding up to <math>t^1</math> gives
<math display="block">P_0(x) = 1 \,,\quad P_1(x) = x.</math>
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 <math>P_n</math>'s without resorting to direct expansion of the [[Taylor series]], however. Equation&nbsp;{{EquationNote|2}} is differentiated with respect to {{mvar|t}} on both sides and rearranged to obtain
<math display="block">\frac{x-t}{\sqrt{1-2xt+t^2}} = \left(1-2xt+t^2\right) \sum_{n=1}^\infty n P_n(x) t^{n-1} \,.</math>
Replacing the quotient of the square root with its definition in Eq.&nbsp;{{EquationNote|2}}, and [[equating the coefficients]] of powers of {{math|''t''}} in the resulting expansion gives ''Bonnet’s recursion formula''
<math display="block"> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,.</math>
This relation, along with the first two polynomials {{math|''P''<sub>0</sub>}} and {{math|''P''<sub>1</sub>}}, allows all the rest to be generated recursively.

The generating function approach is directly connected to the [[multipole expansion]] in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782.