orthopolynom (version 1.0-6.1)

hermite.he.polynomials: Create list of Hermite polynomials

Description

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

Usage

hermite.he.polynomials(n, normalized=FALSE)

Value

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

1

order 0 Hermite polynomial

2

order 1 Hermite polynomial

...

n+1

order \(n\) Hermite polynomial

Arguments

n

integer value for thehighest polynomial order

normalized

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

Author

Frederick Novomestky

Details

The function hermite.he.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

hermite.he.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

###
### gemerate a list of normalized Hermite polynomials of orders 0 to 10
###
normalized.p.list <- hermite.he.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Hermite polynomials of orders 0 to 10
###
unnormalized.p.list <- hermite.he.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )