| ultraspherical.polynomials {orthopolynom} | R Documentation |
Create list of ultraspherical polynomials
Description
This function returns a list with n + 1 elements containing
the order k ultraspherical polynomials, C_k^{\left( \alpha \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
ultraspherical.polynomials(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
polynomial parameter |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function ultraspherical.recurrences produces a data frame with the recurrence relation parameters
for the polynomials. If the normalized argument is FALSE, the
function orthogonal.polynomials is used to construct the list of orthogonal polynomial objects.
Otherwise, the function orthonormal.polynomials is used to construct the
list of orthonormal polynomial objects.
Value
A list of n + 1 polynomial objects
1 |
order 0 ultraspherical polynomial |
2 |
order 1 ultraspherical polynomial |
...
n+1 |
order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.
Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
gegenbauer.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized ultra spherical polynomials
### of orders 0 to 10
###
normalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized ultra spherical polynomials
### of orders 0 to 10
###
unnormalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=FALSE )
print( unnormalized.p.list )