Built-in Mathematica Symbol

FactorialPower

FactorialPower[x, n]
gives the factorial power

.

FactorialPower[x, n, h]
gives the step-h factorial power

.

MORE INFORMATION

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 .

FactorialPower is to Sum and DifferenceDelta what Power is to Integrate and D.

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 (2)

Find the "factorial square" of 10:

FactorialPower does not automatically expand out:

Use FunctionExpand to do the expansion:

Find the "factorial square" of 10:

In[1]:=

Out[1]=

FactorialPower does not automatically expand out:

In[1]:=

Out[1]=

Use FunctionExpand to do the expansion:

In[2]:=

Out[2]=

Scope (5)

FactorialPower works with any numbers, not just integers:

Evaluate to arbitrary precision:

The precision of the output tracks the precision of the input:

FactorialPower threads element-wise over lists:

FactorialPower can be expressed in terms of gamma functions:

TraditionalForm formatting:

Generalizations & Extensions (2)

With step -1, 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 (5)

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:

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

: