gives the Lucas number .


gives the Lucas polynomial .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The satisfy the recurrence relation with , .
  • For any complex value of n the are given by the general formula , where is the golden ratio.
  • The Lucas polynomial is the coefficient of in the expansion of .
  • The Lucas polynomials satisfy the recurrence relation .
  • LucasL can be evaluated to arbitrary numerical precision.
  • LucasL automatically threads over lists.
  • LucasL can be used with Interval and CenteredInterval objects. »


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Basic Examples  (4)

Compute Lucas numbers:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (38)

Numerical Evaluation  (5)

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:

LucasL can be used with Interval and CenteredInterval objects:

Specific Values  (6)

Values of LucasL at fixed points:

LucasL for symbolic n:

Values at zero:

Find the value of x in which LucasL[2,x]=5:

Compute the associated LucasL[7,x] polynomial:

Compute the associated LucasL[1/2,x] polynomial for half-integer n:

Visualization  (4)

Plot the LucasL polynomial for various orders:

Plot the real part of TemplateBox[{2}, LucasL](z):

Plot the imaginary part of TemplateBox[{2}, LucasL](z):

Plot as real parts of two parameters vary:

Types 2 and 3 of LucasL function have different branch cut structures:

Function Properties  (14)

LucasL is defined for all real and complex values:

The range of TemplateBox[{n, x}, LucasL2] is all real numbers for odd :

Its range over the complex plane is all complex numbers for any natural number :

Lucas polynomial of an odd order is odd:

Lucas polynomial of an even order is even:

LucasL has the mirror property TemplateBox[{n, {z, }}, LucasL2]=TemplateBox[{n, z}, LucasL2]:

LucasL threads elementwise over lists:

TemplateBox[{n, x}, LucasL2] is an analytic function of :

LucasL is neither non-decreasing nor non-increasing for even values:

LucasL is non-decreasing for odd values:

LucasL is not injective for even values:

LucasL is not surjective for even values:

LucasL is non-negative for even values:

LucasL does not have singularity nor discontinuity:

LucasL is convex for even values:

TraditionalForm formatting:

Differentiation  (3)

First derivatives with respect to n:

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when n=4:

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

Series Expansions  (4)

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:

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Lucas numbers are related to the Fibonacci numbers by the identities:

The ordinary generating function of LucasL:

Generalizations & Extensions  (1)

Lucas polynomials:

Applications  (6)

Solve the Fibonacci recurrence equation:

Find ratios of successive Lucas numbers:

Compare with continued fractions:

Convergence to the Golden Ratio:

Calculate the number of ways to write an integer as a sum of Lucas numbers :

Plot the counts for the first hundred integers:

Find the first Lucas number above 1000000:

First few Lucas pseudoprimes:

Compute Artin's constant:

Properties & Relations  (10)

Expand in terms of elementary functions:

Limiting ratio:

Explicit recursive definition:

Simplify some expressions involving Lucas numbers:

Generating function:

Extract Lucas numbers as coefficients:

LucasL can be represented as a DifferenceRoot:

General term in the series expansion of LucasL:

The generating function for LucasL:

FindSequenceFunction can recognize the LucasL sequence:

The exponential generating function for LucasL:

Possible Issues  (2)

Large arguments can give results too large to be computed explicitly:

Results for integer arguments may not hold for non-integers:

Neat Examples  (2)

Wolfram Research (2007), LucasL, Wolfram Language function, (updated 2008).


Wolfram Research (2007), LucasL, Wolfram Language function, (updated 2008).


Wolfram Language. 2007. "LucasL." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008.


Wolfram Language. (2007). LucasL. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_lucasl, author="Wolfram Research", title="{LucasL}", year="2008", howpublished="\url{}", note=[Accessed: 18-July-2024 ]}


@online{reference.wolfram_2024_lucasl, organization={Wolfram Research}, title={LucasL}, year={2008}, url={}, note=[Accessed: 18-July-2024 ]}