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
1inner product of order 0 orthogonal polynomial
2inner product of order 1 orthogonal polynomial
...
n+1inner 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.
###### 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 )