orthopolynom (version 1.0-3)

hermite.h.inner.products: Inner products of Hermite polynomials

Description

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

Usage

hermite.h.inner.products(n)

Arguments

n

integer value for highest polynomial order

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

Details

The formula used to compute the innner product is as follows. $h_n = \left\langle {H_n |H_n } \right\rangle = \sqrt \pi \;2^n \;n!$.

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
### Hermite polynomials of orders 0 to 10
###
h <- hermite.h.inner.products( 10 )
print( h )