7.12 The Pochhammer Notation

The Pochhammer notation (a)_k (also called Pochhammer’s symbol) is supported by the binary operator Pochhammer(a,k). For a non-negative integer k, it is defined as (http://dlmf.nist.gov/5.2.iii)

(a)0	= 1,
(a)k	= a(a + 1)(a + 2)(a + k − 1).
For a≠0,−1,−2,−3…, this is equivalent to
(a)k	=

When n is integral, the defining product is expanded (assuming the switch EXP is ON). With ROUNDED off, this expression is evaluated numerically if a is numerical and k is integral, and otherwise may be simplified where appropriate. The simplification rules are based upon algorithms supplied by Wolfram Koepf [Koe92].

The Pochammer symbol is used quite extensively in the simplification and numerical evaluation of special functions.