gives the factorial power .
gives the step-h factorial power .
- 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. »
Examplesopen allclose all
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:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
FactorialPower can be used with Interval and CenteredInterval objects:
Specific Values (5)
Values of FactorialPower at fixed points:
FactorialPower for symbolic n:
Find a value of x for which FactorialPower[x,1/7]=1.2:
Evaluate the associated FactorialPower[x,4] polynomial for integer n:
Plot the FactorialPower function for various orders:
Plot FactorialPower as a function of its parameter :
Function Properties (11)
Real domain of the double factorial:
Function range of FactorialPower:
FactorialPower threads elementwise over lists:
is neither non-decreasing nor non-increasing:
FactorialPower is neither non-negative nor non-positive:
has potential singularities and discontinuities when is a negative integer:
is neither convex nor concave:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first two approximations around :
Generalizations & Extensions (2)
With step , FactorialPower gives the rising factorial:
FactorialPower can be applied to a power series:
The number of triples of distinct digits:
Approximate a function using Newton's forward difference formula [MathWorld]:
Construct an approximation by truncating the series:
Properties & Relations (7)
FactorialPower is to Sum as Power is to Integrate:
FactorialPower satisfies :
FactorialPower can always be expressed as a ratio of gamma functions:
FactorialPower[x,x] is equivalent to x!:
The rising factorial is equivalent to a Pochhammer symbol:
The generating function for FactorialPower:
The exponential generating function for FactorialPower:
Possible Issues (2)
Generically, Power is recovered as a limit of of FactorialPower:
This may not be true, however, if is kept on the negative real axis:
Generic series expansion around the origin may not be defined at integer points:
Use assumptions to refine the result:
Wolfram Research (2008), FactorialPower, Wolfram Language function, 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.
Wolfram Language. (2008). FactorialPower. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorialPower.html