Legendre Polynomials |

**In General** \ s.a. Special Functions.

$ __Def__: The solutions
of the equation (think of *x* as cos *θ*),

\[\def\dd{{\rm d}}{\dd\over\dd x}\,\Big[(1-x^2)\,{\dd P_n\over\dd x}\Big] + n(n+1)\,P_n(x) = 0\;.\]

* __From a generating function__:
The Rodrigues formula, *P*_{n}(*x*)
= (1/2^{n}*n*!)
d^{n}/d*x*^{n}
(*x*^{2}−1)^{n}.

* __Orthonormality__: Expressed by the relationship

∫_{−1}^{1}
d*x* *P*_{n}(*x*)
*P*_{m}(*x*)
= 2/(2*n*+1) δ_{nm} .

* __Examples__:

*P*_{0}(*x*)
= 1,* P*_{1}(*x*)
= *x*, *P*_{2}(*x*)
= (3*x*^{2}−1)/2,
*P*_{3}(*x*)
= (5*x*^{3}−3*x*)/2.

* __Properties__:
*P*_{n}(1)
= 1, *P*_{n}(−1)
= (−1)^{n},
*P*_{n}(0)
= 0 for *n* odd.

* __Recursion relations__:
*P'*_{n+1}
= (2*n*+1) *P*_{n}
+ *P'*_{n−1};
(*n*+1) *P*_{n+1}
+ *n* *P*_{n−1}
= (2*n*+1) *xP*_{n} .

**Associated Legendre Functions**

$ __Def__: The solutions of the equation

\[{\dd\over\dd x}\,\Big[(1-x^2)\,{\dd P_{nm}\over\dd x}\Big] + \Big[n(n+1)-{m^2\over(1-x^2)}\Big]\,P_{nm}(x) = 0\;.\]

* __Generating function__:

*P*_{nm}(*x*)
= (−1)^{m}/(2^{n}*n*!)
(1−*x*^{2})^{m/2}
{d^{n+m}/d*x*^{n+m})
(*x*^{2}−1)^{n} =
(−1)^{m}
(1−*x*^{2})^{m/2}
(d^{m}/d*x*^{m})
*P*_{n}(*x*) .

@ __References__: Saharian JPA(09)-a0904 [summation formula].

**References**

@ __General__: in Wyld 76.

@ __Modified__: Yang et al mp/02 [deformed];
Durand JMP(03)mp/02 [fractional operators].

@ __Related topics__: Khusnutdinov JMP(03)mp [uniform asymptotic expansion].

> __Online resources__:
see Wikipedia page.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 17 jan 2016