orthopolynom (version 1.0-5)

gegenbauer.inner.products: Inner products of Gegenbauer polynomials

Description

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

Usage

gegenbauer.inner.products(n,alpha)

Arguments

n

integer value for the highest polynomial order

alpha

numeric value for the polynomial parameter

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 inner products is as follows.

\(h_n = \left\langle {C_n^{\left( \alpha \right)} |C_n^{\left( \alpha \right)} } \right\rangle = \left\{ {\begin{array}{*{20}c} {\frac{{\pi \;2^{1 - 2\,\alpha } \,\Gamma \left( {n + 2\,\alpha } \right)}} {{n!\;\left( {n + \alpha } \right)\,\left[ {\Gamma \left( \alpha \right)} \right]^2 }}} & {\alpha \ne 0} \\ {\frac{{2\;\pi }} {{n^2 }}} & {\alpha = 0} \\

\end{array} } \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.

See Also

ultraspherical.inner.products

Examples

# NOT RUN {
###
### generate the inner products vector for the 
### Gegenbauer polynomials of orders 0 to 10
### the polynomial parameter is 1.0
###
h <- gegenbauer.inner.products( 10, 1 )
print( h )
# }