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The pochhammer symbol is defined for a positive integer n and complex number z as

This is extended analytically to complex $n$ by using the formula

At all points $\left(z\,a\right)$ such that $z$ and $z+a$ are positive integers, this is equivalent to:

In the case that $z$ is a non-positive integer, pochhammer(z,a) is defined by this limit.

In the case that both $z$ and $z+a$ are non-positive integers, Maple also signals the invalid_operation numeric event, allowing the user to control this singular behavior by catching the event. See numeric_events for more information.

$\mathrm{pochhammer}\left(5\,3\right)$

$210$

$\mathrm{pochhammer}\left(z\,2\right)$

$\mathrm{pochhammer}\left(z\,2\right)$

$\mathrm{pochhammer}\left(z\,-3\right)$

$\frac{1}{\mathrm{pochhammer}\left(-3+z\,3\right)}$

$\mathrm{pochhammer}\left(2\,\mathrm{I}\right)$

$\mathrm{\Gamma }\left(2+\mathrm{I}\right)$

$\mathrm{NumericStatus}\left(\mathrm{invalid\_operation}=\mathrm{false}\right)\:$

$\mathrm{pochhammer}\left(-3\,2\right)$

$6$

$\mathrm{NumericStatus}\left(\mathrm{invalid\_operation}=\mathrm{false}\right)$

$\mathrm{invalid\_operation}=\mathrm{true}$

$\mathrm{pochhammer}\left(0\,0\right)$

$1$

$\mathrm{NumericStatus}\left(\mathrm{invalid\_operation}\right)$

$\mathrm{true}$

$\mathrm{diff}\left(\mathrm{pochhammer}\left(a\,x\right)\,x\right)$

$\mathrm{pochhammer}\left(a\,x\right)\mathrm{\Psi }\left(x+a\right)$

$\mathrm{diff}\left(\mathrm{pochhammer}\left(a\,x\right)\,a\right)$

$\mathrm{pochhammer}\left(a\,x\right)\left(\mathrm{\Psi }\left(x+a\right)-\mathrm{\Psi }\left(a\right)\right)$

$\mathrm{series}\left(\mathrm{pochhammer}\left(a\,x\right)\,x\,3\right)$

$1+\mathrm{\Psi }\left(a\right)x+\left(\frac{\mathrm{\Psi }\left(1\,a\right)}{2}+\frac{{\mathrm{\Psi }\left(a\right)}^2}{2}\right)x^2+O\left(x^3\right)$

$\mathrm{pochhammer}\left(x\,5\right)$

$\mathrm{pochhammer}\left(x\,5\right)$

$\mathrm{expand}\left(\right)$

$x^5+10x^4+35x^3+50x^2+24x$

$\mathrm{pochhammer}\left(2\,\frac{1}{3}\right)$

$\frac{8\mathrm{\pi }\sqrt{3}}{27\mathrm{\Gamma }\left(\frac{2}{3}\right)}$

$\mathrm{evalf}\left(\right)$

$1.190639350$

$\mathrm{pochhammer}\left(-3.7+2.2\mathrm{I}\,1.5+2.7\mathrm{I}\right)$

$\mathrm{-0.0005620896042}+0.01961129135\mathrm{I}$

$\mathrm{convert}\left(\mathrm{pochhammer}\left(a\,x\right)\,\mathrm{\Gamma }\right)$

$\frac{\mathrm{\Gamma }\left(x+a\right)}{\mathrm{\Gamma }\left(a\right)}$

$\mathrm{convert}\left(\mathrm{pochhammer}\left(a\,x\right)\,\mathrm{binomial}\right)$

$\left(\genfrac{}{}{0ex}{}{a+x-1}{a-1}\right)x!$

$\mathrm{convert}\left(\mathrm{pochhammer}\left(a\,x\right)\,\mathrm{factorial}\right)$

$\frac{\left(a+x-1\right)!}{\left(a-1\right)!}$