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I'm working on a problem where I need to solve for b in terms of a and z, given $z = (a + b)!$. An approximate solution using the Stirling Approximation: $z = \sqrt {2\pi (a + b)}\left(\frac {(a + b)}{e}\right)^{(a + b)}$would suffice but I'm having trouble with the algebra and Wolfram seems to run out of compute time before generating a solution for me.

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A solution is $b=\Gamma^{-1}(z)-a-1$ with the (functional) inverse of the Gamma function. If this looks like cheating or you have no access to the $\Gamma^{-1},$ you have to solve it numerically, e.g. with Newton's method. The (incomplete) inverse Gamma function is used e.g. in statistics.

For a numerical approximation see Is there an Inverse Gamma $\Gamma^{-1} (z) $ function? , for an approximation of $\Gamma^{-1}$ in terms of LambertW see https://mathoverflow.net/a/28977

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