orthopolynom (version 1.0-6.1)

jacobi.p.inner.products: 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)

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 value for the highest polynomial order

alpha

numeric value for the first polynomial parameter

beta

numeric value for the first polynomial parameter

Author

Frederick Novomestky

Details

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

\(h_n = \left\langle {P_n^{\left( {\alpha ,\beta } \right)} |P_n^{\left( {\alpha ,\beta } \right)} } \right\rangle = \frac{{2^{\alpha + \beta + 1} }} {{2\,n + \alpha + \beta + 1}}\frac{{\Gamma \left( {n + \alpha + 1} \right)\,\Gamma \left( {n + \beta + 1} \right)}} {{n!\;\Gamma \left( {n + \alpha + \beta + 1} \right)}}\).

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 )