Statements (32)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
orthogonal polynomials
|
| gptkbp:alsoKnownAs |
gptkb:Gegenbauer_polynomials
|
| gptkbp:C_0^{(lambda)}(x) |
1
|
| gptkbp:C_1^{(lambda)}(x) |
2 lambda x
|
| gptkbp:definedIn |
interval [-1, 1]
|
| gptkbp:degree |
n
|
| gptkbp:differential |
(1-x^2)y'' - (2lambda+1)xy' + n(n+2lambda)y = 0
|
| gptkbp:domain |
real numbers
|
| gptkbp:hasSpecialCase |
gptkb:Jacobi_polynomials
Legendre polynomials (lambda=1/2) Chebyshev polynomials of the first kind (lambda=0.5) Chebyshev polynomials of the second kind (lambda=1) |
| https://www.w3.org/2000/01/rdf-schema#label |
ultraspherical polynomials
|
| gptkbp:hypergeometricRepresentation |
C_n^{(lambda)}(x) = (2lambda)_n / n! 2F1(-n, 2lambda+n; lambda+1/2; (1-x)/2)
|
| gptkbp:introduced |
gptkb:Leopold_Gegenbauer
|
| gptkbp:namedAfter |
gptkb:Leopold_Gegenbauer
|
| gptkbp:orthogonalityCondition |
integral from -1 to 1 of (1-x^2)^{lambda-1/2} C_m^{(lambda)}(x) C_n^{(lambda)}(x) dx = 0 for m ≠ n
|
| gptkbp:orthogonalOn |
[-1,1]
|
| gptkbp:orthogonalWithRespectTo |
weight function (1-x^2)^{lambda-1/2}
|
| gptkbp:parameter |
lambda
|
| gptkbp:recurrence |
C_{n+1}^{(lambda)}(x) = 2x(n+lambda)/ (n+1) C_n^{(lambda)}(x) - (n+2lambda-1)/(n+1) C_{n-1}^{(lambda)}(x)
|
| gptkbp:satisfies |
partial differential equations
recurrence relation |
| gptkbp:sequence |
C_n^{(lambda)}(x)
|
| gptkbp:usedIn |
harmonic analysis
mathematical physics approximation theory potential theory solution of Laplace's equation in hyperspherical coordinates expansion of functions in series |
| gptkbp:bfsParent |
gptkb:Gegenbauer_polynomials
|
| gptkbp:bfsLayer |
8
|