Associated Laguerre Polynomial Lₙᵅ(x) Calculator

Enter degree n, parameter alpha, and x to evaluate the associated (generalized) Laguerre polynomial Lₙᵅ(x) via a stable three-term recurrence. Shows neighbors, derivative, coefficients, and a plot.

Input

Enter the degree n, parameter alpha, and x to evaluate the associated (generalized) Laguerre polynomial Lₙᵅ(x) using a three-term recurrence.

Degree n

Integer of 0 or more

Parameter alpha

Real number (alpha = 0 gives ordinary Laguerre)

Variable x

Any real number

Result

Lₙᵅ(x) for n = 3, alpha = 1

-1.3333333333

Value at x = 2

Degree n

Parameter alpha

Lₙ₋₁ᵅ for n = 3

-1

Lₙ₊₁ᵅ for n = 3

-1

Derivative of Lₙᵅ for n = 3

-0

Graph of Lₙᵅ(x) for n = 3

The orange dot marks the value at the entered x.

Coefficients of Lₙᵅ(x) for n = 3, alpha = 1

Coefficients for each power of x, listed from the highest degree.

Power of x	Coefficient
3	-0.16666667
2	2
1	-6
0	4

How it works

The associated (generalized) Laguerre polynomial Lₙᵅ(x) is computed with the three-term recurrence (n+1)Lₙ₊₁ᵅ(x) = (2n+1+α−x)Lₙᵅ(x) − (n+α)Lₙ₋₁ᵅ(x), starting from L₀ᵅ(x) = 1 and L₁ᵅ(x) = 1+α−x.
When α = 0 it reduces to the ordinary Laguerre polynomial Lₙ(x).
The derivative uses the identity d/dx Lₙᵅ(x) = −Lₙ₋₁^(α+1)(x).
Coefficients are assembled from the closed form Lₙᵅ(x) = Σ (−1)ᵏ C(n+α, n−k) / k! · xᵏ (k from 0 to n), which allows a real-valued α.
With weight w(x) = xᵅ e^(−x) the polynomials are orthogonal on 0 ≤ x, and in quantum mechanics Lₙᵅ(x) appears in the hydrogen atom radial wavefunction.
For large degree n or large x, floating point cancellation can increase the error.

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