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

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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  (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 :

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: 22-December-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: 22-December-2024 ]}