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:

FactorialPower threads elementwise over lists:

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 FactorialPower function using MatrixFunction:

Values of FactorialPower at fixed points:

Obtain the polynomial representation FactorialPower[x,n] for integer values of n:

With step , FactorialPower[x,n,h] gives the rising factorial:

This is equivalent to Pochhammer:

Expand FactorialPower[x,n] for a fixed value of x:

Do the same while adding integer values for the third argument:

Value with second argument zero:

Value with first argument 0 and positive second argument:

Find a value of x for which FactorialPower[x,1/7]=1.2:

Plot the FactorialPower function for various orders:

Plot FactorialPower as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Real domain of the factorial power:

Complex domain:

Function range of FactorialPower[x,n] for various fixed values of n:

is an analytic function of x:

is neither nondecreasing nor nonincreasing:

is not injective:

is surjective:

FactorialPower is neither non-negative nor non-positive:

has potential singularities and discontinuities when is a negative integer:

is neither convex nor concave:

TraditionalForm formatting:

First derivative of with respect to :

First derivative of with respect to :

Higher derivatives of with respect to :

Plot the higher derivatives with respect to x when n=2:

Find the Taylor expansion using Series:

Plots of the first two approximations around :

Taylor expansion at a generic point:

Series expansion at a singular point:

FactorialPower can be applied to a power series:

For positive integers :

Recurrence relation: