orthopolynom (version 1.0-6.1)

ghermite.h.inner.products: Inner products of generalized Hermite polynomials

Description

This function returns a vector with \(n + 1\) elements containing the inner product of an order \(k\) generalized Hermite polynomial, \(H_k^{\left( \mu \right)} \left( x \right)\), with itself (i.e. the norm squared) for orders \(k = 0,\;1,\; \ldots ,\;n \).

Usage

ghermite.h.inner.products(n, mu)

Value

A vector with \(n + 1\) elements

1

inner product of order 0 orthogonal polynomial

2

inner product of order 1 orthogonal polynomial

...

n+1

inner product of order \(n\) orthogonal polynomial

Arguments

n

n integer value for the highest polynomial order

mu

mu polynomial parameter

Author

Frederick Novomestky

Details

The parameter \(\mu\) must be greater than -0.5. The formula used to compute the inner products is as follows.

\(h_n \left( \mu \right) = \left\langle {H_m^{\left( \mu \right)} |H_n^{\left( \mu \right)} } \right\rangle = 2^{2\,n} \,\left[ {\frac{n} {2}} \right]!\;\Gamma \left( {\left[ {\frac{{n + 1}} {2}} \right] + \mu + \frac{1} {2}} \right)\)

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.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

Examples

###
### generate the inner products vector for the
### generalized Hermite polynomials of orders 0 to 10
### polynomial parameter is 1
###
h <- ghermite.h.inner.products( 10, 1 )
print( h )