schebyshev.t.recurrences: Recurrence relations for shifted Chebyshev polynomials

Description

This function returns a data frame with \(n + 1\) rows and four named columns containing the coefficient vectors c, d, e and f of the recurrence relations for the order \(k\) shifted Chebyshev polynomial of the first kind, \(T_k^* \left( x \right)\), and for orders \(k = 0,\;1,\; \ldots ,\;n\).

Usage

schebyshev.t.recurrences(n, normalized)

Arguments

integer value for the highest polynomial order

normalized

boolean value which, if TRUE, returns recurrence relations for normalized polynomials

Value

A data frame with the recurrence relation parameters.

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.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

Examples

Run this code

# NOT RUN {
###
### generate the recurrence relations for 
### the normalized shifted T Chebyshev polynomials
### of orders 0 to 10
###
normalized.r <- schebyshev.t.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for 
### the unnormalized shifted T Chebyshev polynomials
### of orders 0 to 10
###
unnormalized.r <- schebyshev.t.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
# }