Laguerre Polynomial Lₙ(x) Calculator

Enter the degree n and x to compute the Laguerre polynomial Lₙ(x) with a three-term recurrence, plus neighboring degrees, the derivative and the coefficient list.

Input

Compute the Laguerre polynomial Lₙ(x) with a three-term recurrence. Enter the degree n and x.

Degree n

Integer from 0 to 200

Value of x

Any real number

Result

Value of L_4(x)

-0.2890625

at x = 1.5

Degree n

L_(4-1)(x)

-0.6875

L_(4+1)(x)

0.11640625

Derivative L_4'(x)

1.0625

Graph of L_4(x)

Curve over the interval 0 to 10. The orange dot marks the entered x.

Coefficients of L_4(x)

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

Power of x	Coefficient
4	0.04166667
3	-0.66666667
2	3
1	-4
0	1

How it works

The Laguerre polynomials start from L_0(x)=1 and L_1(x)=1−x and are evaluated with the three-term recurrence (n+1)L_(n+1)(x)=(2n+1−x)Lₙ(x)−nL_(n-1)(x).
They are orthogonal on the interval from 0 to infinity with the weight function e^(−x) and appear in quantum mechanics, for example in the radial wavefunctions of the hydrogen atom.
The coefficient list is built with integer arithmetic from the closed form Lₙ(x)=Σ_(k=0..n) (−1)^k C(n,k)/k! x^k.
The derivative is obtained from the differential recurrence x Lₙ'(x)=n(Lₙ(x)−L_(n-1)(x)), using Lₙ'(0)=−n at x=0.
For large degree n and large x the value grows rapidly and floating point rounding error increases, so treat the displayed value as an approximation.
Enter the degree n as an integer from 0 to 200.

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