gives the Laguerre polynomial .


gives the generalized Laguerre polynomial .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given when possible.
  • .
  • The Laguerre polynomials are orthogonal with weight function .
  • They satisfy the differential equation .
  • For certain special arguments, LaguerreL automatically evaluates to exact values.
  • LaguerreL can be evaluated to arbitrary numerical precision.
  • LaguerreL automatically threads over lists.
  • LaguerreL[n,x] is an entire function of x with no branch cut discontinuities.


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Basic Examples  (6)

Compute the 5^(th) LaguerreL:

Compute the associated Laguerre polynomial :

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (31)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (5)

Values of LaguerreL at fixed points:

Values at zero:

Find the first positive minimum of LaguerreL[10,x ]:

Compute the associated LaguerreL[7,x] polynomial:

Different LaguerreL types give different symbolic forms:

Visualization  (3)

Plot the LaguerreL polynomial for various orders:

Plot the real part of TemplateBox[{10}, LucasL](x+iy):

Plot the imaginary part of TemplateBox[{10}, LucasL](x+iy):

Plot as real parts of two parameters vary:

Function Properties  (5)

LaguerreL is defined for all real and complex values:

Approximate function range of LaguerreL:

LaguerreL has the mirror property :

LaguerreL threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when n=3:

Formula for the ^(th) derivative with respect to x:

Integration  (3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

Series Expansions  (5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (3)

LaguerreL may reduce to simpler form:

Generating function of LaguerreL:

Recurrence identity:

Generalizations & Extensions  (1)

LaguerreL can be applied to a power series:

Applications  (4)

Solve the Laguerre differential equation:

Generalized Fourier series for functions defined on :

Radial wave-function of the hydrogen atom:

Compute the energy eigenvalue from the differential equation:

The energy is independent of the orbital quantum number l:

The number of derangement anagrams for a word with character counts :

Count the number of derangements for the word Mathematica:

Direct count:

Properties & Relations  (7)

Get the list of coefficients in a Laguerre polynomial:

Use FunctionExpand to expand LaguerreL functions into simpler functions:

LaguerreL can be represented as a DifferentialRoot:

LaguerreL can be represented in terms of MeijerG:

LaguerreL can be represented as a DifferenceRoot:

General term in the series expansion of LaguerreL:

The generating function for LaguerreL:

Possible Issues  (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Wolfram Research (1988), LaguerreL, Wolfram Language function,


Wolfram Research (1988), LaguerreL, Wolfram Language function,


@misc{reference.wolfram_2020_laguerrel, author="Wolfram Research", title="{LaguerreL}", year="1988", howpublished="\url{}", note=[Accessed: 22-January-2021 ]}


@online{reference.wolfram_2020_laguerrel, organization={Wolfram Research}, title={LaguerreL}, year={1988}, url={}, note=[Accessed: 22-January-2021 ]}


Wolfram Language. 1988. "LaguerreL." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). LaguerreL. Wolfram Language & System Documentation Center. Retrieved from