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 .
  • 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.

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  (22)

Numerical Evaluation  (4)

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:

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 TemplateBox[{{(, {x, +, {i,  , y}}, )}, n}, FactorialPower]:

Plot the imaginary part of TemplateBox[{{(, {x, +, {i,  , y}}, )}, n}, FactorialPower]:

Function Properties  (4)

Real domain of the double factorial:

Complex domain:

Function range of FactorialPower:

FactorialPower threads elementwise over lists:

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  (2)

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:

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 :

Introduced in 2008
 (7.0)