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 $sum_xTemplateBox[{x, k}, FactorialPower]$ 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 (30)

Numerical Evaluation (5)

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 can be used with Interval and CenteredInterval objects:

Specific Values (5)

Values of FactorialPower at fixed points:

FactorialPower for symbolic n:

Values at zero:

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

Evaluate the associated FactorialPower[x,4] polynomial for integer n:

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 (11)

Real domain of the double factorial:

Complex domain:

Function range of FactorialPower:

FactorialPower threads elementwise over lists:

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:

TraditionalForm formatting:

Differentiation (2)

First derivative with respect to n:

First derivative with respect to x:

Higher derivatives with respect to x:

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

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first two approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications (2)

For positive integers $TemplateBox[{x, n}, FactorialPower]= ((-1)^n TemplateBox[{{n, -, x}}, Gamma])/(TemplateBox[{{-, x}}, Gamma])$ :

Recurrence relation:

Generalizations & Extensions (2)

With step , FactorialPower gives the rising factorial:

FactorialPower can be applied to a power series:

Applications (3)

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 (7)

FactorialPower is to Sum as Power is to Integrate:

FactorialPower satisfies :

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

Compare to the expansion of :

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:

Compare to expansion for explicit value of :

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