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std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel

double      assoc_laguerre( unsigned int n, unsigned int m, double x );
float       assoc_laguerre( unsigned int n, unsigned int m, float x );
long double assoc_laguerre( unsigned int n, unsigned int m, long double x );
float       assoc_laguerref( unsigned int n, unsigned int m, float x );
long double assoc_laguerrel( unsigned int n, unsigned int m, long double x );
 (1) (since C++17) 
double      assoc_laguerre( unsigned int n, unsigned int m, IntegralType x );
 (2) (since C++17)
1) Computes the associated Laguerre polynomials of the degree n, order m, and argument x
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

n - the degree of the polymonial, a value of unsigned integer type
m - the order of the polynomial, a value of unsigned integer type
x - the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the associated Laguerre polynomial of x, that is \((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{L}_{n+m}(x)\)(-1)m
dm
dxm
L
n+m(x), is returned (where \(\mathsf{L}_{n+m}(x)\)L
n+m(x) is the unassociated Laguerre polynomial, std::laguerre(n+m, x)).

Error handling

Errors may be reported as specified in math_errhandling.

  • If the argument is NaN, NaN is returned and domain error is not reported
  • If x is negative, a domain error may occur
  • If n or m is greater or equal to 128, the behavior is implementation-defined.

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

The associated Laguerre polynomials are the polynomial solutions of the equation \(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)xy,,
+(m+1-x)y,
+ny = 0.

The first few are:

  • assoc_laguerre(0, m, x) = 1
  • assoc_laguerre(1, m, x) = -x + m + 1
  • assoc_laguerre(2, m, x) =
    1
    2
    [x2
    -2(m+2)x+(m+1)(m+2)]
  • assoc_laguerre(3, m, x) =
    1
    6
    [-x3
    -3(m+3)x2
    -3(m+2)(m+3)x+(m+1)(m+2)(m+3)]

Example

#include <cmath>
#include <iostream>
double L1(unsigned m, double x) { return -x + m + 1; }
double L2(unsigned m, double x) { return 0.5*(x*x-2*(m+2)*x+(m+1)*(m+2)); }
int main()
{
    // spot-checks
    std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n'
              << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n';
}

Output:

10.5=10.5
60.125=60.125

See also

(C++17)(C++17)(C++17)
 Laguerre polynomials
(function)

Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld--A Wolfram Web Resource.

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