orthopolynom (version 1.0-6.1)

jacobi.p.polynomials: Create list of Jacobi polynomials

Description

This function returns a list with \(n + 1\) elements containing the order \(k\) Jacobi polynomials, \(P_k^{\left( {\alpha ,\beta } \right)} \left( x \right)\), for orders \(k = 0,\;1,\; \ldots ,\;n\).

Usage

jacobi.p.polynomials(n, alpha, beta, normalized=FALSE)

Value

A list of \(n + 1\) polynomial objects

1

order 0 Jacobi polynomial

2

order 1 Jacobi polynomial

...

n+1

order \(n\) Chebyshev polynomial

Arguments

n

integer value for the highest polynomial order

alpha

numeric value for the first polynomial parameter

beta

numeric value for the second polynomial parameter

normalized

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

Author

Frederick Novomestky

Details

The function jacobi.p.recurrences produces a data frame with the recurrence relation parameters for the polynomials. If the normalized argument is FALSE, the function orthogonal.polynomials is used to construct the list of orthogonal polynomial objects. Otherwise, the function orthonormal.polynomials is used to construct the list of orthonormal polynomial objects.

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.

See Also

jacobi.p.recurrences, orthogonal.polynomials, orthonormal.polynomials

Examples

###
### gemerate a list of normalized Jacobi P polynomials of orders 0 to 10
### first parameter value a is 2 and second parameter value b is 2
###
normalized.p.list <- jacobi.p.polynomials( 10, 2, 2, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Jacobi P polynomials of orders 0 to 10
### first parameter value a is 2 and second parameter value b is 2
###
unnormalized.p.list <- jacobi.p.polynomials( 10, 2, 2, normalized=FALSE )
print( unnormalized.p.list )