orthopolynom (version 1.0-6.1)

ghermite.h.polynomials: Create list of generalized Hermite polynomials

Description

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

Usage

ghermite.h.polynomials(n, mu, normalized = FALSE)

Value

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

1

order 0 generalized Hermite polynomial

2

order 1 generalized Hermite polynomial

...

n+1

order \(n\) generalized Hermite polynomial

Arguments

n

integer value for the highest polynomial order

mu

numeric value for the polynomial parameter

normalized

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

Author

Frederick Novomestky

Details

The parameter \(\mu\) must be greater than -0.5. The function ghermite.h.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

Alvarez-Nordase, R., M. K. Atakishiyeva and N. M. Atakishiyeva, 2004. A q-extension of the generalized Hermite polynomials with continuous orthogonality property on R, International Journal of Pure and Applied Mathematics, 10(3), 335-347.

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

ghermite.h.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

###
### gemerate a list of normalized generalized Hermite polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
normalized.p.list <- ghermite.h.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized generalized Hermite polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
unnormalized.p.list <- ghermite.h.polynomials( 10, 1, normalized=FALSE )
print( unnormalized.p.list )