Pochhammer
The rising factorial (a)_n = a(a+1)(a+2)…(a+n−1).
Wikipedia
Falling and rising factorialsMathWorldPochhammerSymbolWikidataQ2339261DLMF5.2FungrimRisingFactorialWolfram LanguagePochhammer✓Pochhammer(a, n)the rising factorial Details
- Closed form via the Gamma function:
. , since the rising factorial from 1 just counts up to n. - Negative order gives the falling reciprocal:
. - Dividing by
counts multisets: . - If a is a non-positive integer with
, the product picks up a zero factor and vanishes. - A building block of hypergeometric series, where it appears in both numerator and denominator terms.