In a reputable source I found the formula $\Gamma(z + n) = (z)_n \Gamma(z)$. What does the notation $(z)_n$ signify?
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5Falling factorial – lulu May 14 '22 at 23:51
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However this doesn't seem to be true. For $z = 3$ and $n = 2$, $\Gamma(z + n) = \Gamma(5) = 24$; $(z)_n = (3)_2 = 3 \cdot 2 = 6$; $\Gamma(z) = \Gamma(3) = 2$; yielding $24 = 12$. – CarbonFlambe May 15 '22 at 00:14
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Ok, I think they meant to place the rising factorial $z^{(n)} = z(z + 1) ... (z + (n - 1))$ in place of the falling factorial $(z)_n$. – CarbonFlambe May 15 '22 at 00:21
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1I was just typing that. Yes, the authors meant the rising factorial. – lulu May 15 '22 at 00:22
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Further, the notation here seems to be a total disaster area. The Pochhammer symbol is denoted $(z)_n$ but is defined as the rising factorial. So that must be what the authors meant. See the Mathworld pages for "falling factorial", "rising factorial", and "Pochhammer symbol". – CarbonFlambe May 15 '22 at 00:43