orthopolynom (version 1.0-6.1)

ultraspherical.recurrences: Recurrence relations for ultraspherical 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 \(k\) ultraspherical polynomial, \(C_k^{\left( \alpha \right)} \left( x \right)\), and for orders \(k = 0,\;1,\; \ldots ,\;n\).

Usage

ultraspherical.recurrences(n, alpha, normalized=FALSE)

Value

A data frame with the recurrence relation parameters.

Arguments

n

integer value for the highest polynomial order

alpha

numeric value for the polynomial parameter

normalized

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

Author

Frederick Novomestky

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

ultraspherical.recurrences

Examples

###
### generate the recurrence relations for 
### the normalized ultraspherical polynomials
### of orders 0 to 10
### polynomial parameter value is 1.0
###
normalized.r <- ultraspherical.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for 
### the normalized ultraspherical polynomials
### of orders 0 to 10
### polynomial parameter value is 1.0
###
unnormalized.r <- ultraspherical.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )