Difference between revisions of "Pochhammer"
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| − | The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by | + | The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by |
| − | $$(a)_0 = 1 | + | $$\left\{ \begin{array}{ll} |
| − | + | (a)_0 &= 1 \\ | |
| − | + | (a)_n \equiv a^{\overline{n}} &= \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1). | |
| + | \end{array} \right.$$ | ||
=Properties= | =Properties= | ||
[[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> | [[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> | ||
| + | |||
| + | =Notes= | ||
| + | We are using this symbol to denote the [[rising factorial]] (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the [[falling factorial]] (as Wikipedia does). | ||
=References= | =References= | ||
Revision as of 18:56, 18 December 2016
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$\left\{ \begin{array}{ll} (a)_0 &= 1 \\ (a)_n \equiv a^{\overline{n}} &= \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1). \end{array} \right.$$
Properties
Sum of reciprocal Pochhammer symbols of a fixed exponent
Notes
We are using this symbol to denote the rising factorial (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the falling factorial (as Wikipedia does).
