FactorialPower [x,n]
gives the factorial power TemplateBox[{x, n}, FactorialPower].
FactorialPower [x,n,h]
gives the step-h factorial power TemplateBox[{x, n, h}, FactorialPower3].
FactorialPower
FactorialPower [x,n]
gives the factorial power TemplateBox[{x, n}, FactorialPower].
FactorialPower [x,n,h]
gives the step-h factorial power TemplateBox[{x, n, h}, FactorialPower3].
Details
- Mathematical function, suitable for both symbolic and numeric manipulation.
- For integer n, TemplateBox[{x, n}, FactorialPower] is given by , and TemplateBox[{x, n, h}, FactorialPower3] is given by .
- TemplateBox[{x, n}, FactorialPower] is given for any n by TemplateBox[{{x, +, 1}}, Gamma]/TemplateBox[{{x, -, n, +, 1}}, Gamma].
- TemplateBox[{TemplateBox[{x, k}, FactorialPower], x}, DifferenceDelta2] is given by k TemplateBox[{x, {k, -, 1}}, FactorialPower] and sum_xTemplateBox[{x, k}, FactorialPower] is given by TemplateBox[{x, {k, +, 1}}, FactorialPower]/(k+1).
- 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 all close allBasic 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 (34)
Numerical Evaluation (7)
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 :
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 TemplateBox[{{(, z, )}, 5}, FactorialPower]:
Plot the imaginary part of TemplateBox[{{(, z, )}, 5}, FactorialPower]:
Function Properties (10)
Real domain of the factorial power:
Complex domain:
Function range of FactorialPower [x,n] for various fixed values of n:
TemplateBox[{x, 3}, FactorialPower] is an analytic function of x:
TemplateBox[{x, 3}, FactorialPower] is neither nondecreasing nor nonincreasing:
TemplateBox[{x, 3}, FactorialPower] is not injective:
TemplateBox[{x, 3}, FactorialPower] is surjective:
FactorialPower is neither non-negative nor non-positive:
TemplateBox[{x, y}, FactorialPower] has potential singularities and discontinuities when is a negative integer:
TemplateBox[{x, 3}, FactorialPower] is neither convex nor concave:
TraditionalForm formatting:
Differentiation (3)
First derivative of TemplateBox[{x, n}, FactorialPower] with respect to :
First derivative of TemplateBox[{x, n}, FactorialPower] with respect to :
Higher derivatives of TemplateBox[{x, n}, FactorialPower] 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 TemplateBox[{x, n}, FactorialPower]= ((-1)^n TemplateBox[{{n, -, x}}, Gamma])/(TemplateBox[{{-, x}}, Gamma]):
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 n!TemplateBox[{x, n}, Binomial]:
FactorialPower [x,x] is equivalent to x!:
Pochhammer can be expressed in terms of a single FactorialPower expression:
Verify the identity TemplateBox[{x, k}, Pochhammer]=TemplateBox[{x, k, {-, 1}}, FactorialPower3] 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 :
Related Links
History
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_2025_factorialpower, author="Wolfram Research", title="{FactorialPower}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialPower.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_factorialpower, organization={Wolfram Research}, title={FactorialPower}, year={2008}, url={https://reference.wolfram.com/language/ref/FactorialPower.html}, note=[Accessed: 04-January-2026]}