orthopolynom (version 1.0-6.1)

ultraspherical.polynomials: 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)

Value

A list of \(n + 1\) polynomial objects

1

order 0 ultraspherical polynomial

2

order 1 ultraspherical polynomial

...

n+1

order \(n\) ultraspherical polynomial

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

Author

Frederick Novomestky

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.

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 )