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

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< cpp‎ | numeric‎ | special math

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

Run this code

    #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

laguerre laguerref laguerrel (C++17)(C++17)(C++17) | Laguerre polynomials (function)

### External links

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

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