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)
Arguments
n
n integer value for the highest polynomial order
mu
mu polynomial parameter
Value
A vector with $n + 1$ elements
1inner product of order 0 orthogonal polynomial
2inner product of order 1 orthogonal polynomial
...
n+1inner product of order $n$ orthogonal polynomial
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.
###### 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 )