Reciprocal Gamma $1/\Gamma(z)$ is an entire function and so it has a convergent Taylor series expansion which was given in its wikipedia article.


The problem is there is no citation for it so I want to ask where I can find a proof/derivation of its power series expansion?

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    $\begingroup$ It follows easily from the expansion for $\ln\Gamma$ which in turn is immediately derived from the infinite product for $\Gamma$, as explained in several references at the end of the Wikipedia article. $\endgroup$ – მამუკა ჯიბლაძე Jun 21 '14 at 2:51

A good derivation of the Gamma function and it's equivalences is Rainville's Special Functions book, see Chapter 2. In there Rainville establishes the equivalent forms of the Gamma functions and relates them to other functions as well. We should also point out that,

$$\frac{1}{\Gamma(z)} = z + \gamma z^2 + \left(\frac{\gamma^2}{2} - \frac{\pi^2}{12}\right)z^3 + \cdots,$$

is as much of the Taylor series as is known, since closed forms of $\zeta(n)$ are only known for he even integers $n$. See here for a recursive formula for the coefficients of the Gamma function's reciprocal.


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