orthopolynom (version 1.0-6.1)

glaguerre.inner.products: Inner products of generalized Laguerre polynomials

Description

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

Usage

glaguerre.inner.products(n,alpha)

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

integer highest polynomial order

alpha

polynomial parameter

Author

Frederick Novomestky

Details

The formula used to compute the inner products is as follows.

\(h_n = \left\langle {L_n^{\left( \alpha \right)} |L_n^{\left( \alpha \right)} } \right\rangle = \frac{{\Gamma \left( {\alpha + n + 1} \right)}} {{n!}}\).

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
### generalized Laguerre polynomial inner products of orders 0 to 10
### polynomial parameter is 1.
###
h <- glaguerre.inner.products( 10, 1 )
print( h )