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

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float assoc_laguerre ( unsigned int n, unsigned int m, float x ); double assoc_laguerre ( unsigned int n, unsigned int m, double x ); long double assoc_laguerre ( unsigned int n, unsigned int m, long double x ); |  | (since C++17) (until C++23)
/* floating-point-type */ assoc_laguerre( unsigned int n, unsigned int m, /* floating-point-type */ x ); |  | (since C++23)
float assoc_laguerref( unsigned int n, unsigned int m, float x ); | (2) | (since C++17)
long double assoc_laguerrel( unsigned int n, unsigned int m, long double x ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double assoc_laguerre ( unsigned int n, unsigned int m, Integer x ); | (A) | (since C++17)

1-3) Computes the associated Laguerre polynomials of the degree n, order m, and argument x. The library provides overloads of `std::assoc_laguerre` for all cv-unqualified floating-point types as the type of the parameter x. (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

n | - | the degree of the polynomial, an unsigned integer value
m | - | the order of the polynomial, an unsigned integer value
x | - | the argument, a floating-point or integer value

### 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:

Function | Polynomial
--- | ---
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)]

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::assoc_laguerre(int_num1, int_num2, num) has the same effect as std::assoc_laguerre(int_num1, int_num2, static_cast<double>(num)).

### 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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