FactorialPower
✖
FactorialPower
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
open allclose allBasic Examples (7)Summary of the most common use cases
Find the "factorial square" of 10:

https://wolfram.com/xid/0puvkna6ym-evw7dj

FactorialPower does not automatically expand out:

https://wolfram.com/xid/0puvkna6ym-d09b2d

Use FunctionExpand to do the expansion:

https://wolfram.com/xid/0puvkna6ym-f8pd0e

Plot over a subset of the reals:

https://wolfram.com/xid/0puvkna6ym-gi13sj

Plot over a subset of complexes:

https://wolfram.com/xid/0puvkna6ym-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0puvkna6ym-ej957w

Series expansion at Infinity:

https://wolfram.com/xid/0puvkna6ym-cugjvu

Series expansion at a singular point:

https://wolfram.com/xid/0puvkna6ym-ii6s9p

Scope (34)Survey of the scope of standard use cases
Numerical Evaluation (7)

https://wolfram.com/xid/0puvkna6ym-l274ju


https://wolfram.com/xid/0puvkna6ym-c0cnf


https://wolfram.com/xid/0puvkna6ym-b0wt9

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0puvkna6ym-y7k4a


https://wolfram.com/xid/0puvkna6ym-hfml09

Evaluate efficiently at high precision:

https://wolfram.com/xid/0puvkna6ym-di5gcr


https://wolfram.com/xid/0puvkna6ym-bq2c6r

FactorialPower threads elementwise over lists:

https://wolfram.com/xid/0puvkna6ym-be0lrf

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

https://wolfram.com/xid/0puvkna6ym-h0d6g


https://wolfram.com/xid/0puvkna6ym-dj6d9x

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0puvkna6ym-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0puvkna6ym-thgd2

Or compute the matrix FactorialPower function using MatrixFunction:

https://wolfram.com/xid/0puvkna6ym-o5jpo

Specific Values (6)
Values of FactorialPower at fixed points:

https://wolfram.com/xid/0puvkna6ym-nww7l

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

https://wolfram.com/xid/0puvkna6ym-hfz8z6

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

https://wolfram.com/xid/0puvkna6ym-hllsms

This is equivalent to Pochhammer:

https://wolfram.com/xid/0puvkna6ym-gtnb6s

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

https://wolfram.com/xid/0puvkna6ym-mxd8v8

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

https://wolfram.com/xid/0puvkna6ym-brla0c

Value with second argument zero:

https://wolfram.com/xid/0puvkna6ym-2ew296

Value with first argument 0 and positive second argument:

https://wolfram.com/xid/0puvkna6ym-e41pf2

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

https://wolfram.com/xid/0puvkna6ym-f2hrld


https://wolfram.com/xid/0puvkna6ym-jk6twr

Visualization (3)
Plot the FactorialPower function for various orders:

https://wolfram.com/xid/0puvkna6ym-bhh92c

Plot FactorialPower as a function of its parameter :

https://wolfram.com/xid/0puvkna6ym-bzo8ib


https://wolfram.com/xid/0puvkna6ym-ouu484


https://wolfram.com/xid/0puvkna6ym-k1fl0c

Function Properties (10)
Real domain of the factorial power:

https://wolfram.com/xid/0puvkna6ym-cl7ele


https://wolfram.com/xid/0puvkna6ym-de3irc

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

https://wolfram.com/xid/0puvkna6ym-fphbrc


https://wolfram.com/xid/0puvkna6ym-gva6yl

is neither nondecreasing nor nonincreasing:

https://wolfram.com/xid/0puvkna6ym-2ra8g


https://wolfram.com/xid/0puvkna6ym-c9npzh


https://wolfram.com/xid/0puvkna6ym-b5buvp


https://wolfram.com/xid/0puvkna6ym-patce


https://wolfram.com/xid/0puvkna6ym-bcrbvs

FactorialPower is neither non-negative nor non-positive:

https://wolfram.com/xid/0puvkna6ym-dvzykj

has potential singularities and discontinuities when
is a negative integer:

https://wolfram.com/xid/0puvkna6ym-b9vmv1


https://wolfram.com/xid/0puvkna6ym-wr28ur

is neither convex nor concave:

https://wolfram.com/xid/0puvkna6ym-l0srvu

TraditionalForm formatting:

https://wolfram.com/xid/0puvkna6ym-eksj3l


https://wolfram.com/xid/0puvkna6ym-lo9x7a

Differentiation (3)
First derivative of with respect to
:

https://wolfram.com/xid/0puvkna6ym-dzs0u

First derivative of with respect to
:

https://wolfram.com/xid/0puvkna6ym-krpoah

Higher derivatives of with respect to
:

https://wolfram.com/xid/0puvkna6ym-z33jv

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

https://wolfram.com/xid/0puvkna6ym-fxwmfc

Series Expansions (3)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0puvkna6ym-ewr1h8

Plots of the first two approximations around :

https://wolfram.com/xid/0puvkna6ym-binhar

Taylor expansion at a generic point:

https://wolfram.com/xid/0puvkna6ym-jwxla7

FactorialPower can be applied to a power series:

https://wolfram.com/xid/0puvkna6ym-i2ilw0

Applications (4)Sample problems that can be solved with this function
The number of length-r permutations of a length-n list of distinct elements is given by FactorialPower[n,r]:

https://wolfram.com/xid/0puvkna6ym-cqomie


https://wolfram.com/xid/0puvkna6ym-utr67f

The number of triples of distinct digits:

https://wolfram.com/xid/0puvkna6ym-e4r7y7


https://wolfram.com/xid/0puvkna6ym-juzo07

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

https://wolfram.com/xid/0puvkna6ym-dx32nb

Construct an approximation by truncating the series:

https://wolfram.com/xid/0puvkna6ym-b5bsu4

https://wolfram.com/xid/0puvkna6ym-bicv3j


https://wolfram.com/xid/0puvkna6ym-bgrlpq

Compare with their integral definition:

https://wolfram.com/xid/0puvkna6ym-f4ze40

Properties & Relations (11)Properties of the function, and connections to other functions
FactorialPower is to Sum as Power is to Integrate:

https://wolfram.com/xid/0puvkna6ym-p2ftcl


https://wolfram.com/xid/0puvkna6ym-kfzqkv

FactorialPower satisfies :

https://wolfram.com/xid/0puvkna6ym-fpg6oj

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

https://wolfram.com/xid/0puvkna6ym-cwpcq2

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

https://wolfram.com/xid/0puvkna6ym-bbobbj

Compare with the expansion of :

https://wolfram.com/xid/0puvkna6ym-bjrx1e

FactorialPower[x,n] is equivalent to :

https://wolfram.com/xid/0puvkna6ym-nv31v

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

https://wolfram.com/xid/0puvkna6ym-crecf9


https://wolfram.com/xid/0puvkna6ym-gjrxez

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

https://wolfram.com/xid/0puvkna6ym-bou6h

Verify the identity for integer
:

https://wolfram.com/xid/0puvkna6ym-y077no

This function is often called the rising factorial:

https://wolfram.com/xid/0puvkna6ym-omot2s

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

https://wolfram.com/xid/0puvkna6ym-zq9r2

FactorialPower can be represented as a DifferenceRoot:

https://wolfram.com/xid/0puvkna6ym-ojcks5


https://wolfram.com/xid/0puvkna6ym-d7mw27

The generating function for FactorialPower:

https://wolfram.com/xid/0puvkna6ym-pz93yz

The exponential generating function for FactorialPower:

https://wolfram.com/xid/0puvkna6ym-gaiyeu

Possible Issues (2)Common pitfalls and unexpected behavior
Generically, Power is recovered as the limit as of FactorialPower:

https://wolfram.com/xid/0puvkna6ym-b0oh28

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

https://wolfram.com/xid/0puvkna6ym-joqyv


https://wolfram.com/xid/0puvkna6ym-e101dt

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

https://wolfram.com/xid/0puvkna6ym-flxb51


https://wolfram.com/xid/0puvkna6ym-ghsky1

Use assumptions to refine the result:

https://wolfram.com/xid/0puvkna6ym-c8s3gp


https://wolfram.com/xid/0puvkna6ym-jpclz

Compare with the expansion for an explicit value of :

https://wolfram.com/xid/0puvkna6ym-izwwp6

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.
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.
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
Wolfram Language. (2008). FactorialPower. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorialPower.html
BibTeX
@misc{reference.wolfram_2025_factorialpower, author="Wolfram Research", title="{FactorialPower}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialPower.html}", note=[Accessed: 13-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_factorialpower, organization={Wolfram Research}, title={FactorialPower}, year={2008}, url={https://reference.wolfram.com/language/ref/FactorialPower.html}, note=[Accessed: 13-May-2025
]}