Legendre polynomials (aka. Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
are solutions of the Legendre differential equation].
The Legendre polynomial, can be defined by the contour integral
The first seven Legendre polynomials are:
"shifted" Legendre polynomials (which obey the orthogonality relationship):
Powers in terms of Legendre polynomials:
Associated Legendre polynomials.
etc.