orthopolynom (version 1.0-6.1)

ghermite.h.recurrences: Recurrence relations for generalized Hermite 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 \(k\) generalized Hermite polynomial, \(H_k^{\left( \mu \right)} \left( x \right)\), and for orders \(k = 0,\;1,\; \ldots ,\;n\).

Usage

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

Value

A data frame with the recurrence relation parameters.

Arguments

n

integer value for the highest polynomial order

mu

numeric value for the polynomial parameter

normalized

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.

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.inner.products

Examples

###
### generate the recurrences data frame for 
### the normalized generalized Hermite polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
normalized.r <- ghermite.h.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for 
### the unnormalized generalized Hermite polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
unnormalized.r <- ghermite.h.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )