Legendre Polynomial of the First Kind, $P_\nu(x)$

Computes the Legendre polynomial of the first kind, $P_\nu(x)$, for real order $\nu$ (integer or fractional) and real argument $x$.

Usage

Pn(n, x)

Arguments

n: Numeric vector. Degree (order) of the Legendre polynomial. Can be integer or fractional (e.g., 0, 1, 2.5, 3.7). Can also be negative.
x: Numeric vector. Real argument(s) at which to evaluate the polynomial. Valid for all real values including $|x| > 1$.

Value

A numeric matrix of dimension length(n) by length(x), where element [i, j] contains $P_{n_i}(x_j)$.

Details

The Legendre polynomial of the first kind satisfies the differential equation: $$(1 - x^2) \frac{d^2 P_\nu}{dx^2} - 2x \frac{dP_\nu}{dx} + \nu(\nu + 1) P_\nu = 0$$

For integer order $n$, the function uses the recurrence relation: $$P_0(x) = 1$$ $$P_1(x) = x$$ $$P_n(x) = \frac{(2n-1) x P_{n-1}(x) - (n-1) P_{n-2}(x)}{n}$$

For fractional order $\nu$, the function uses:

For $|x| \leq 1$: The Ferrers function via the hypergeometric series $P_\nu(x) = {}_2F_1(-\nu, \nu+1; 1; \frac{1-x}{2})$
For $|x| > 1$: A numerical contour integral representation

Note

This function calls underlying $C++$ code via Rcpp for computational efficiency and to support different cases for both order and argument that are not readily available in R.

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications. Chapter 8: Legendre Functions.

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/14

Examples

# Single values
Pn(1, 1)
#>      [,1]
#> [1,]    1

# Multiple orders, single argument
Pn(c(1, 2, 3), 1)
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    1

# Single order, multiple arguments
Pn(1, c(1, 2, 3))
#>      [,1] [,2] [,3]
#> [1,]    1    2    3

# Multiple orders and arguments (returns a matrix)
Pn(c(1, 2, 3), c(1, 2, 3))
#>      [,1] [,2] [,3]
#> [1,]    1  2.0    3
#> [2,]    1  5.5   13
#> [3,]    1 17.0   63

# Fractional orders
Pn(c(0.5, 2.2, 3), c(1, 2, 3))
#>      [,1]      [,2]      [,3]
#> [1,]    1  1.329138  1.597387
#> [2,]    1  6.849800 17.720229
#> [3,]    1 17.000000 63.000000

# Negative arguments
Pn(c(0.5, 2, -3), c(1, -2.5, 3))
#>      [,1]         [,2]      [,3]
#> [1,]    1 8.994435e-17  1.597387
#> [2,]    1 8.875000e+00 13.000000
#> [3,]    0 0.000000e+00  0.000000

Legendre Polynomial of the First Kind, \(P_\nu(x)\)