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:
“The “female culture” has shifted more rapidly than the “male culture”; the image of the go-get ‘em woman has yet to be fully matched by the image of the let’s take-care-of-the-kids- together man. More important, over the last thirty years, men’s underlying feelings about taking responsibility at home have changed much less than women’s feelings have changed about forging some kind of identity at work.”
—Arlie Hochschild (20th century)