Pochhammer

The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(a)_n = \dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes gamma.

Properties

Sum of reciprocal Pochhammer symbols of a fixed exponent
Pochhammer symbol with non-negative integer subscript

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

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.1 (2)$
• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $5.1 (3)$
• 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 $1.1$