LucasL
Details

- 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.
Examples
open allclose allBasic Examples (4)
Scope (30)
Numerical Evaluation (4)
Specific Values (6)
Visualization (4)
Function Properties (7)
LucasL is defined for all real and complex values:
Approximate function range of LucasL:
Lucas polynomial of an odd order is odd:
Lucas polynomial of an even order is even:
LucasL has the mirror property :
LucasL threads elementwise over lists:
TraditionalForm formatting:
Differentiation (3)
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:
Function Identities and Simplifications (2)
Lucas numbers are related to the Fibonacci numbers by the identities:
The ordinary generating function of LucasL:
Applications (5)
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:
Properties & Relations (10)
Expand in terms of elementary functions:
Explicit recursive definition:
Simplify some expressions involving Lucas numbers:
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)
Text
Wolfram Research (2007), LucasL, Wolfram Language function, https://reference.wolfram.com/language/ref/LucasL.html (updated 2008).
BibTeX
BibLaTeX
CMS
Wolfram Language. 2007. "LucasL." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/LucasL.html.
APA
Wolfram Language. (2007). LucasL. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LucasL.html