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Legendre Polynomials

(useful sequence of polynomials)

The Legendre Polynomials are a sequence of ever-longer polynomials, one of each non-negative-integer polynomial degree, in a pattern analogous to the pattern of Binomial Coefficients, but more complicated. One representation:

  • P0(x) = 1
  • P1(x) = x
  • Pn+1(x) = ( (2n+1)xPn(x)-nPn-1(x) ) / (n+1)

where Pn(x) is the Legendre polynomial to the degree of n. They occur in the series solutions (in the manner of a Taylor Series) to some useful differential equations, including for:

(Items in this list of applications are not necessarily distinct from each other.)