orthopolynom (version 1.0-2)

ultraspherical.polynomials: Create list of ultraspherical polynomials

Description

This function returns a list with $n$+1 elements containing the order $k$ ultraspherical polynomials, $C_k^{\left( \alpha \right)} \left( x \right)$, for orders $k$ = 0, 1, ..., $n$.

Usage

ultraspherical.polynomials(n, alpha, normalized=FALSE)

Arguments

n

integer highest polynomial order

alpha

polynomial parameter

normalized

a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials

Value

• A list of $n$+1 polynomial objects
• 1
  order 0 ultraspherical polynomial
• 2
  order 1 ultraspherical polynomial
• ...
• n+1
  order $n$ ultraspherical polynomial

Details

The function produces a data frame with the recurrence relation parameters for the orthogonal polynomials. It then uses the function orthogonal.polynomials to construct the list of polynomial objects from the recurrence relations.

References

Abramowitz and Stegun (1968)

See Also

gegenbauer.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

normalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=TRUE )
unnormalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=FALSE )