I have come across a moment generating function for a random variable $Y$ of the following form $$M_Y(t) = \mathbb E\left[e^{tY}\right] = \left[1 - \frac t \beta\right]^k.$$ So it is basically $[M_X(t)]^{-1}$ for some $X$ that is $\text{Gamma}(k, \beta)$ distributed. Is there a known distribution for $Y$?

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    $\begingroup$ Inverting the corresponding characteristic function indicates the original is a "generalized distribution" rather than a legitimate probability distribution. How did you obtain this mgf? Perhaps a mistake was made in its derivation. $\endgroup$ – whuber Feb 22 '18 at 17:47
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    $\begingroup$ Thanks a lot for that. I don't think it was a mistake. I was calculating the MGF of a Stochastic Differential Equation with known MGF but with added noise in the drift. The result was a product of terms of the known MGF and the MGF that I have above. Can you clarify what you mean by "generalized distribution," as I know there is a generalized distribution for the Gamma? $\endgroup$ – Jack Feb 23 '18 at 9:41
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    $\begingroup$ See en.wikipedia.org/wiki/Distribution_(mathematics). $\endgroup$ – whuber Feb 23 '18 at 14:58

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