The Laguerre polynomials are orthogonal polynomials with respect to the weighting function on the half-line . They are denoted by the letter “” with the order as subscript and are normalized by the condition that the coefficient of the highest order term of is .
The first few Laguerre poynomials are as follows:
A generalization is given by the associated Laguerre polynomials which depends on a parameter (traditionally denoted “”). As it turns out, they are polynomials of the argument as as well, so they are polynomials of two variables. They are defined over the same interval with the same normalization condition, but the weight function is generalized to . They are notated by including the parameter as a parenthesized superscript (not all authors use the parentheses).
The ordinary Laguerre polynomials are the special case of the generalized Laguerre polynomials when the parameter goes to zero. When some result holds for generalized Laguerre polynomials which is not more complicated than that for ordinary Laguerre polynomials, we shall only provide the more general result and leave it to the reader to send the parameter to zero to recover the more specific result.
The first few generalized Laguerre polynomials are as follows:
2 Formulae for these polynomials
They can be computed from a Rodrigues formula:
3 Equations they satisfy
The Laguerre polynomials satisfy the orthogonality relation
The Laguerre polynomials satisfy the differential equation
This equation arises in many contexts such as in the quantum mechanical treatment of the hydrogen atom.
|Date of creation||2013-03-22 12:30:56|
|Last modified on||2013-03-22 12:30:56|
|Last modified by||rspuzio (6075)|
|Defines||associated Laguerre polynomial|