# Laguerre Differential Equation & Polynomials

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## Laguerre Differential Equation: Definition

The Laguerre differential equation (sometimes just called the Laguerre equation) has the format (Dettman, 1986):

xy′′ + (1 – x)y′ + λy = 0

Where λ is a real number (n or k are sometimes used as notation instead of λ).

This ordinary differential equation is usually only of interest when λ is an integer and the interval is x ≥ 0 (Tenenbaum & Pollard, 1985, p.624). It has multiple applications in the study of optics and harmonic oscillators in quantum mechanics; The wave function for the hydrogen atom is the Laguerre differential equation.

In standard form, the equation becomes (Konstantogiannis, 2018):

There is a regular singular point at x = 0 and an irregular singularity at ∞.

The equation and associated polynomials are named after Edmond Laguerre (1834-1866).

## Laguerre Polynomials

The solutions to the Laguerre differential equation are the Laguerre polynomials (Glasgow, 2014). They can be expressed as a series (Joseph, 2019):

Or, alternatively by Rodrigue’s formula (Mahmoudi et al., 2012):

They can also be defined recursively by:

For example, the next few Laguerre polynomials are:

• l2 = ½ (x2 – 4x + 2),
• l3 = (1/6) (-x3 + 9x2 – 18x + 6).
• l4 = (1/24) (x4 – 16x3 + 72x2 – 96x + 24).
• l5 = (1/120) (-x5 + 25x4 – 200x3 + 600x2 – 600x + 120).
• l6 = (1/720) (x6 – 36x5 + 450x4 – 2400x3 + 5400x2 – 4320x + 720).

Plot of the first three Laguerre polynomials,

## References

Dettman, J. (1986). Introduction to Linear Algebra and Differential Equations. Dover.
Glasgow, L. (2014). Applied Mathematics for Science and Engineering. Wiley.
Joseph, A. (2019). Phy310—Mathematical Methods for Physicists I. Retrieved August 25, 2020 from: https://web.iisermohali.ac.in/Faculty/anoshjoseph/courses/2019_odd_mmp1/lec12.pdf
Konstantogiannis, S. (2018). Polynomial and Non-Polynomial Terminating Series Solutions to the Associated Laguerre Differential Equation. Retrieved August 25, 20202 from: https://www.academia.edu/37908752/Polynomial_and_Non_Polynomial_Terminating_Series_Solutions_to_the_Associated_Laguerre_Differential_Equation
Mahmoudi, Y. et al. (2012). Adomian Decomposition Method with Laguerre Polynomials for Solving Ordinary Differential Equation. Retrieved August 25, 2020 from: https://www.textroad.com/pdf/JBASR/J.%20Basic.%20Appl.%20Sci.%20Res.,%202(12)12236-12241,%202012.pdf
Tenenbaum, M. & Pollard, H. (1985). Ordinary differential equations : an elementary textbook for students of mathematics, engineering, and the sciences. Dover Publications.

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Stephanie Glen. "Laguerre Differential Equation & Polynomials" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/laguerre-differential-equation-polynomials/
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