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Write $\mu =\alpha /\gamma$ and $\varphi =1/\alpha$. With $\theta =-\gamma /\alpha =-1/\mu <0$, the Gamma exponential dispersion family representation iswhereSubstitution gives the original gamma density, so both unknown parameters are included. The exponential-family derivative identities yield $\mu =b^{\prime }(\theta )=-1/\theta$ and $b^{\prime \prime }(\theta )=1/{\theta }^2={\mu }^2$. ThereforeThe exact natural-parameter canonical link function is $g_{\mathrm{can}}(\mu )=-1/\mu$. The usual inverse-link convention is $g(\mu )=1/\mu$, as used by the printed R fits; multiplying the link and coefficients by $-1$ gives the natural-parameter convention. The sign convention changes neither the fitted means nor the weight matrix below.

An exponential dispersion family has densities of the formwhere $\theta$ is the canonical parameter, $\varphi >0$ is the dispersion parameter, and the support does not depend on $\theta$. Its mean and variance are

For the given Gamma distribution, setThengivesThus the gamma laws form the Gamma exponential dispersion family.

Directly integrating, or using the cumulant-generating function of a gamma distribution,soDifferentiation at zero yields