orthopolynom (version 1.0-6.1)

ghermite.h.weight: Weight function for the generalized Hermite polynomial

Description

This function returns the value of the weight function for the order \(k\) generalized Hermite polynomial, \(H_k^{\left( \mu \right)} \left( x \right) \).

Usage

ghermite.h.weight(x, mu)

Value

The value of the weight function

Arguments

x

a numeric vector function argument

mu

polynomial parameter

Author

Frederick Novomestky

Details

The function takes on non-zero values in the interval \( \left( -\infty,\infty \right) \). The parameter \(\mu\) must be greater than -0.5. The formula used to compute the generalized Hermite weight function is as follows.

\(w\left( {x,\mu } \right) = \left| x \right|^{2\;\mu } \;\exp \left( { - x^2 } \right)\)

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

###
### compute the generalized Hermite weight function for argument values 
### between -3 and 3
###
x <- seq( -3, 3, .01 )
y <- ghermite.h.weight( x, 1 )