orthopolynom (version 1.0-6.1)

schebyshev.u.polynomials: Create list of shifted Chebyshev polynomials

Description

This function returns a list with \(n + 1\) elements containing the order \(k\) shifted Chebyshev polynomials of the second kind, \(U_k^* \left( x\right)\), for orders \(k = 0,\;1,\; \ldots ,\;n\).

Usage

schebyshev.u.polynomials(n, normalized)

Value

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

1

order 0 shifted Chebyshev polynomial

2

order 1 shifted Chebyshev polynomial

...

n+1

order \(n\) shifted Chebyshev polynomial

Arguments

n

integer value for highest polynomial order

normalized

a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials

Author

Frederick Novomestky

Details

The function schebyshev.u.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

schebyshev.u.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

###
### gemerate a list of normalized shifted U Chebyshev polynomials of orders 0 to 10
###
normalized.p.list <- schebyshev.u.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized shifted U Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- schebyshev.u.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )