FactorialPower

The falling factorial x(x-1)…(x-n+1), n factors, for a concrete-number x.

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FactorialPower(x, n)x(x-1)(x-2)…(x-n+1)

FactorialPower(x, n, h)x(x-h)(x-2h)…(x-(n-1)h), stepping by h instead of 1

Typed signatures

Type	Package	Applies
(number, number, number?) -> number	combinatorics	always

Domain: Collections

Details

• A negative integer n inverts the product -- FactorialPower(x, -m, h) = 1 / ((x+h)(x+2h)…(x+mh)).
• A non-integer n (step h = 1 only) generalizes via Gamma(x+1)/Gamma(x-n+1).
• A symbolic x is left unevaluated -- Wolfram itself only opens the product via FunctionExpand, not by default.

Examples

Basic #

A symbolic x stays unevaluated, matching Wolfram's own default behavior

live ↗

In$\mathrm{FactorialPower}(x, 3)$Out$\mathrm{FactorialPower}(x, 3)$

5 × 4 × 3 = 60

live ↗

In$\mathrm{FactorialPower}(5, 3)$Out$60$

Scope #

A symbolic x with a third (step) argument also stays unevaluated

live ↗

In$\mathrm{FactorialPower}(x, 2, h)$Out$\mathrm{FactorialPower}(x, 2, h)$

The empty product is 1, even for a symbolic x

live ↗

In$\mathrm{FactorialPower}(x, 0)$Out$1$

About

Examples	4
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