generalized Laguerre polynomials
GPTKB entity
Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
orthogonal polynomials
|
| gptkbp:alsoKnownAs |
associated Laguerre polynomials
|
| gptkbp:appearsIn |
mathematical physics
numerical analysis approximation theory |
| gptkbp:definedIn |
L_n^{(α)}(x)
|
| gptkbp:domain |
real numbers
|
| gptkbp:firstPolynomial |
L_0^{(α)}(x) = 1
|
| gptkbp:generatingFunction |
(1-t)^{-α-1} exp(-xt/(1-t)) = Σ_{n=0}^∞ L_n^{(α)}(x) t^n
|
| gptkbp:hasSpecialCase |
gptkb:Laguerre_polynomials
|
| https://www.w3.org/2000/01/rdf-schema#label |
generalized Laguerre polynomials
|
| gptkbp:namedAfter |
gptkb:Edmond_Laguerre
|
| gptkbp:orthogonalOn |
[0, ∞)
|
| gptkbp:orthogonalWithRespectTo |
weight function x^α e^{-x} on [0, ∞)
|
| gptkbp:parameter |
α
n |
| gptkbp:recurrence |
(n+1)L_{n+1}^{(α)}(x) = (2n+α+1-x)L_n^{(α)}(x) - (n+α)L_{n-1}^{(α)}(x)
|
| gptkbp:satisfies |
second-order linear differential equation
|
| gptkbp:secondPolynomial |
L_1^{(α)}(x) = -x + α + 1
|
| gptkbp:usedIn |
quantum mechanics
radial part of hydrogen atom wavefunctions |
| gptkbp:bfsParent |
gptkb:Laguerre_polynomials
|
| gptkbp:bfsLayer |
7
|