I'm trying to find the motivation behind the extended form of the exponential family of distributions in the fundamental paper on GLM by Nelder and Wedderburn (Generalized Linear Models, J. R. Statist. Soc. A (1972), 135, Part 3, p. 370).
In section 1.1 they simply jump right in, and state that they will be dealing with distributions of the form
But how did they come up with the phi-dependence in this form? What was the motivation for introducing alpha(phi) and beta(phi,z)? Did they just stare at a bunch of distributions that contain a phi-like parameter, and then infer this form? If so, can someone point me to an exhaustive list of the distributions they stared at? :) Or did they use some other considerations for introducing alpha(phi) and beta(phi,z)?
Any insights would be greatly appreciated!