Derivative of the Legendre Polynomial of the First Kind

Computes the $k$-th derivative of the Legendre polynomial of the first kind, $\frac{d^k}{dx^k} P_\nu(x)$, with respect to the argument $x$.

Usage

Pndk(n, x, k = 1L)

Arguments

n: Numeric vector. Degree (order) of the Legendre polynomial. Can be integer or fractional.
x: Numeric vector. Real argument(s) at which to evaluate the derivative.
k: Integer. Order of the derivative ($k \geq 0$). Default is 1 for the first derivative.

Value

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

Details

For integer order $n$:

The derivative is computed using the relationship with associated Legendre polynomials: $$\frac{d^m P_n(x)}{dx^m} = \frac{1}{(1-x^2)^{m/2}} P_n^m(x)$$

where $P_n^m(x)$ is the associated Legendre polynomial (the Boost C++ uses Condon-Shortley phase convention).

At the endpoints $x = \pm 1$, the known closed-form expressions are used: $$P_n^{(k)}(1) = \frac{1}{2^k k!} \prod_{j=0}^{k-1} (n-j)(n+1+j)$$ $$P_n^{(k)}(-1) = (-1)^{n+k} P_n^{(k)}(1)$$

If $k > n$ for integer $n$, the result is 0 (derivative of a polynomial of degree $n$ taken more than $n$ times).

For fractional order $\nu$:

Derivatives are computed using central finite differences: $$\frac{dP_\nu}{dx} \approx \frac{P_\nu(x+h) - P_\nu(x-h)}{2h}$$

Higher-order derivatives use the generalized finite difference stencil. Note that accuracy may be limited for fractional orders due to the numerical integration underlying Pn.

Note

For fractional orders, the finite difference approximation may have reduced accuracy (typically 4-6 significant digits) compared to integer orders. A warning is issued for higher-order derivatives ($k > 1$) of fractional orders.

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. Dover Publications. Section 8.5: Associated Legendre Functions.

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/14.10

Examples

# First derivative of P_2(x) at x = 0.5
# P_2(x) = (3x^2 - 1)/2, so P'_2(x) = 3x, P'_2(0.5) = 1.5
Pndk(2, 0.5, 1)
#>      [,1]
#> [1,]  1.5

# Second derivative of P_2(x)
# P''_2(x) = 3
Pndk(2, 0.5, 2)
#>      [,1]
#> [1,]    3

# Multiple orders
Pndk(c(1, 2, 3), 0.5, 1)
#>       [,1]
#> [1,] 1.000
#> [2,] 1.500
#> [3,] 0.375

# First derivative at multiple points
Pndk(2, c(-0.5, 0, 0.5), 1)
#>      [,1] [,2] [,3]
#> [1,] -1.5    0  1.5

# Fractional order (uses finite differences)
Pndk(0.5, 0.5, 1)
#>           [,1]
#> [1,] 0.4503717

# Derivative at endpoint
# P'_n(1) = n(n+1)/2
Pndk(3, 1, 1)  # Should be 3*4/2 = 6
#>      [,1]
#> [1,]    6