Legendre Polynomials - Shifted Legendre Polynomials

The shifted Legendre polynomials are defined as . Here the "shifting" function (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval to the interval, implying that the polynomials are orthogonal on :

An explicit expression for the shifted Legendre polynomials is given by

The analogue of Rodrigues' formula for the shifted Legendre polynomials is

The first few shifted Legendre polynomials are:

n
0 1
1
2
3

Read more about this topic:  Legendre Polynomials

Famous quotes containing the word shifted:

    “They shifted a little, but not
    to return my stare.”
    —Elizabeth Bishop (1911–1979)