| ghermite.h.inner.products {orthopolynom} | R Documentation |
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)
Arguments
n |
|
mu |
|
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)
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 |
Author(s)
Frederick Novomestky fnovomes@poly.edu
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 )