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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 (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 :

Plot the imaginary part of :

Function Properties (10)

Real domain of the factorial power:

Complex domain:

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

is an analytic function of x:

is neither nondecreasing nor nonincreasing:

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 (3)

First derivative of with respect to :

First derivative of with respect to :

Higher derivatives of 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 :

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

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

Verify the identity 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 :

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