Exponential Family Representation: Dumb question on scale parameter and whether it went to Hawaii

So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale).

https://projecteuclid.org/euclid.ss/1177013604 $$p(x;\theta,\phi) = \exp \left\{ \frac{[\theta \cdot x−A(\theta)]} {a(\phi)} +c(x,\phi) \right\}$$ where $$\cdot$$ is a dot product and $$c$$ and $$a$$ are functions.

Whereas I am more familiar with the location only representation - see for example Definition of exponential family

$$p(x;\theta) = h(x) \exp( \theta \cdot T(x)−A(\theta))$$ where $$\cdot$$ is a dot product. What is going on? its likely the difference between density and distribution with the phi parameters defining the normalizing constant.

Would like some details or a reference would be most appreciated. Observations:

1. The $$c(x,\phi)$$ is $$c(x)$$ or basically $$h(x) = \exp(c(x))$$ and $$a(\phi) =1$$. No idea why x became T(x)
2. Note the derivative of the cumulant $$A(\theta)$$ in each equation is equal to the expected value of x and T(x) respectively. WHich indicates the two are not equivalent.
• It would help with posterity if you could write out the equations yourself in $\LaTeX$ so others will be able to get something out of the question in case a link breaks. – Taylor Nov 27 '18 at 22:57
• Latex is super rusty..will try and further update once I know what to do. – pythOnometrist Nov 27 '18 at 23:05
• Hopefully that will help a bit. Make sure I got everything right – Taylor Nov 27 '18 at 23:08
• A handy Mathjax reference can be found here: math.meta.stackexchange.com/questions/5020/… in case you ever need a hint! – Sycorax Nov 28 '18 at 0:51