Knowlpedia

Definition

Rising factorial

The product a(a+1)...(a+n-1), including the empty product at n=0.

For a∈Ca\in\mathbb C and a nonnegative integer nn, the rising factorial is

(a)0=1,(a)n=∏j=0n−1(a+j).(a)_0=1,\qquad (a)_n=\prod_{j=0}^{n-1}(a+j).

The same notation is also called the Pochhammer symbol; some conventions use an upward arrow to distinguish it from a falling factorial. It satisfies

(a)n+1=(a+n)(a)n,(a)m+n=(a)m(a+m)n.(a)_{n+1}=(a+n)(a)_n,\qquad (a)_{m+n}=(a)_m(a+m)_n.
Examples

(1)n=n!(1)_n=n!, whereas (a)2=a(a+1)(a)_2=a(a+1). The definition is a polynomial in aa; it therefore remains meaningful at parameters where a quotient of gamma functions used to represent it would contain poles.