# hermite.h.recurrences: Recurrence relations for Hermite polynomials In orthopolynom: Collection of functions for orthogonal and orthonormal polynomials

## Description

This function returns a data frame with n + 1 rows and four named columns containing the coefficient vectors c, d, e and f of the recurrence relations for the order k Hermite polynomial, H_k ≤ft( x \right), and for orders k = 0,\;1,\; … ,\;n.

## Usage

 1 hermite.h.recurrences(n, normalized=FALSE) 

## Arguments

 n integer value for the highest polynomial order normalized boolean value which, if TRUE, returns recurrence relations for normalized polynomials

## Value

A data frame with the recurrence relation parameters.

## 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.

hermite.h.inner.products,

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 ### ### generate the recurrences data frame for ### the normalized Hermite H polynomials ### of orders 0 to 10. ### normalized.r <- hermite.h.recurrences( 10, normalized=TRUE ) print( normalized.r ) ### ### generate the recurrences data frame for ### the unnormalized Hermite H polynomials ### of orders 0 to 10. ### unnormalized.r <- hermite.h.recurrences( 10, normalized=FALSE ) print( unnormalized.r ) 

### Example output

Loading required package: polynom
c d         e         f
1  1 0 1.4142136 0.0000000
2  1 0 1.0000000 0.7071068
3  1 0 0.8164966 0.8164966
4  1 0 0.7071068 0.8660254
5  1 0 0.6324555 0.8944272
6  1 0 0.5773503 0.9128709
7  1 0 0.5345225 0.9258201
8  1 0 0.5000000 0.9354143
9  1 0 0.4714045 0.9428090
10 1 0 0.4472136 0.9486833
11 1 0 0.4264014 0.9534626
c d e  f
1  1 0 2  0
2  1 0 2  2
3  1 0 2  4
4  1 0 2  6
5  1 0 2  8
6  1 0 2 10
7  1 0 2 12
8  1 0 2 14
9  1 0 2 16
10 1 0 2 18
11 1 0 2 20


orthopolynom documentation built on May 2, 2019, 3:22 a.m.