Difference between revisions of "Pochhammer"
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The Pochhammer symbol $(a)_n$ 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 | ||
| − | $$\left\{ \begin{array}{ll} | + | $$(a)_n = \left\{ \begin{array}{ll} |
| − | + | 1, & \quad n=0 \\ | |
| − | + | \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1), & \quad n=1,2,3,\ldots. | |
\end{array} \right.$$ | \end{array} \right.$$ | ||
Revision as of 19:07, 17 June 2017
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(a)_n = \left\{ \begin{array}{ll} 1, & \quad n=0 \\ \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1), & \quad n=1,2,3,\ldots. \end{array} \right.$$
Properties
Sum of reciprocal Pochhammer symbols of a fixed exponent
Relationship between Pochhammer and gamma
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
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent formula $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
- 1964: W.N. Bailey: Generalized Hypergeometric Series ... (next): Section $1.1$
