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7.13 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)

\begin{align*} (a)_0 &= 1, \\ (a)_k &= a(a+1)(a+2)\cdots (a+k-1). \end{align*}

For \(a \neq 0, -1, -2, -3, \ldots \), this is equivalent to

\[ (a)_k = \frac {\Gamma (a+k)}{\Gamma (a)}. \]
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 Pochhammer symbol is used quite extensively in the simplification and numerical evaluation of special functions.

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