orthopolynom (version 1.0-6.1)

slegendre.polynomials: Create list of shifted Legendre polynomials

Description

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

Usage

slegendre.polynomials(n, normalized=FALSE)

Value

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

1

order 0 shifted Legendre polynomial

2

order 1 shifted Legendre polynomial

...

n+1

order \(n\) shifted Legendre polynomial

Arguments

n

integer value for the highest polynomial order

normalized

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

Author

Frederick Novomestky

Details

The function slegendre.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

slegendre.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

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