/C++

# 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