# FactorialPower

FactorialPower[x,n]

gives the factorial power .

FactorialPower[x,n,h]

gives the step-h factorial power .

# Details

• Mathematical function, suitable for both symbolic and numeric manipulation.
• For integer n, is given by , and is given by .
• is given for any n by .
• is given by and is given by .
• FactorialPower[x,n] evaluates automatically only when x and n are numbers.
• FunctionExpand always converts FactorialPower to a polynomial or combination of gamma functions.
• FactorialPower can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(7)

Find the "factorial square" of 10:

FactorialPower does not automatically expand out:

Use FunctionExpand to do the expansion:

Plot over a subset of the reals:

Plot over a subset of complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(33)

### Numerical Evaluation(6)

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:

FactorialPower can be used with Interval and CenteredInterval objects:

### Specific Values(6)

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:

### Visualization(3)

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 :

### Function Properties(10)

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 non-decreasing nor non-increasing:

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:

### Differentiation(3)

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:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plots of the first two approximations around :

Taylor expansion at a generic point:

FactorialPower can be applied to a power series:

### Function Identities and Simplifications(2)

For positive integers :

Recurrence relation:

## Applications(4)

The number of length-r permutations of a length-n list of distinct elements is given by FactorialPower[n,r]:

The number of triples of distinct digits:

Approximate a function using Newton's forward difference formula [MathWorld]:

Construct an approximation by truncating the series:

First 10 Nørlund numbers:

Compare with their integral definition:

## Properties & Relations(11)

FactorialPower is to Sum as Power is to Integrate:

FactorialPower satisfies :

This makes FactorialPower analogous to Power and its relationship to D:

FactorialPower can always be expressed as a ratio of gamma functions:

Compare with the expansion of :

FactorialPower[x,n] is equivalent to :

FactorialPower[x,x] is equivalent to x!:

Pochhammer can be expressed in terms of a single FactorialPower expression:

Verify the identity for integer :

This function is often called the rising factorial:

Verify an expansion of FactorialPower in terms of Pochhammer for the first few cases:

FactorialPower can be represented as a DifferenceRoot:

The generating function for FactorialPower:

The exponential generating function for FactorialPower:

## Possible Issues(2)

Generically, Power is recovered as the limit as of FactorialPower:

This may not be true, however, if is kept on the negative real axis:

The generic series expansion around the origin may not be defined at integer points:

Use assumptions to refine the result:

Compare with the expansion for an explicit value of :

Wolfram Research (2008), FactorialPower, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialPower.html.

#### Text

Wolfram Research (2008), FactorialPower, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialPower.html.

#### CMS

Wolfram Language. 2008. "FactorialPower." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FactorialPower.html.

#### APA

Wolfram Language. (2008). FactorialPower. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorialPower.html

#### BibTeX

@misc{reference.wolfram_2024_factorialpower, author="Wolfram Research", title="{FactorialPower}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialPower.html}", note=[Accessed: 18-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_factorialpower, organization={Wolfram Research}, title={FactorialPower}, year={2008}, url={https://reference.wolfram.com/language/ref/FactorialPower.html}, note=[Accessed: 18-July-2024 ]}