# Definition of exponential family with dispersion parameter

I was recently reading a discussion of generalized linear models that considered the response to come from an exponential family with a dispersion parameter so $$f(y|\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)} + c(y, \phi)\right)$$ for known function $$a$$, $$b$$, and $$c$$. Everywhere else the official definition of an exponential family that I've seen is $$g(y|\theta) = \exp(\theta^T T(y) - A(\theta))h(y).$$

How do I reconcile these two definitions? Is $$f$$ really something slightly different?

I could define a new parameter $$\xi = \theta / a(\phi)$$ and then I have $$f(y) = \exp\left(y\xi - d(\theta,\phi)\right)c^*(y, \phi)$$ with $$d(\theta,\phi) = b(\theta) / a(\phi)$$ and $$c^* = \exp \circ c$$ but I've got $$\phi$$ potentially interacting with the support which isn't ok if it's supposed to be a parameter, and the partition function $$d$$ can depend on $$\theta$$ and $$\phi$$ separately rather than just on $$\xi$$. So that doesn't seem to make them coincide. What's going on here?

The definition you quote which is used with generalized linear models (glm) is not an exponential family, it is an exponential dispersion family. For a fixed value of the dispersion parameter $$\phi$$ it is an exponential family (indexed by $$\theta$$), but when $$\phi$$ varies it is not.
When used in glm's, the exponential dispersion family is used for inference about $$\theta$$, but eventual inference about the dispersion parameter $$\phi$$ is done outside that framework.