ultraspherical.inner.products: Inner products of ultraspherical polynomials

In orthopolynom: Collection of Functions for Orthogonal and Orthonormal Polynomials

Inner products of ultraspherical polynomials

Description

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

Usage

ultraspherical.inner.products(n,alpha)

Arguments

n	integer value for the highest polynomial order
alpha	numeric value for the polynomial parameter

Details

This function uses the same formula as the function gegenbauer.inner.products.

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

Author(s)

Frederick Novomestky [e-mail]

References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., NY.

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

gegenbauer.inner.products

Examples

###
### generate the inner products vector for the
### ultraspherical polynomials of orders 0 to 10.
### the polynomial parameter is 1.0
###
h <- ultraspherical.inner.products( 10, 1 )
print( h )

orthopolynom documentation built on Oct. 3, 2022, 5:08 p.m.