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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix LucasL function using MatrixFunction:

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:

Plot the LucasL polynomial for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

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

LucasL is defined for all real and complex values:

The range of 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 :

LucasL threads elementwise over lists:

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:

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 derivative with respect to x:

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:

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

The ordinary generating function of LucasL: