| jacobi.p.inner.products {orthopolynom} | R Documentation |
Inner products of Jacobi polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Jacobi polynomial, P_k^{\left( {\alpha ,\beta } \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
jacobi.p.inner.products(n,alpha,beta)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the first polynomial parameter |
beta |
numeric value for the first polynomial parameter |
Details
The formula used to compute the innser products is as follows.
h_n = \left\langle {P_n^{\left( {\alpha ,\beta } \right)} |P_n^{\left( {\alpha ,\beta } \right)} } \right\rangle = \frac{{2^{\alpha + \beta + 1} }}
{{2\,n + \alpha + \beta + 1}}\frac{{\Gamma \left( {n + \alpha + 1} \right)\,\Gamma \left( {n + \beta + 1} \right)}}
{{n!\;\Gamma \left( {n + \alpha + \beta + 1} \right)}}.
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 |
Author(s)
Frederick Novomestky fnovomes@poly.edu
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 product vector for the P Jacobi polynomials of orders 0 to 10
###
h <- jacobi.p.inner.products( 10, 2, 2 )
print( h )