If an exponential family is given by:
$g(y|\theta) = exp\{\theta^TT(y)-A(\theta)\}h(y)$
then the functions $h(y)$, $A(\theta)$ and $T(y)$ are defined by names:
$T(y)$ is a sufficient statistic
$A(\theta)$ is a cumulant function
$h(y)$ is an underlying measure
For an exponential dispersion family:
$f_Y(y;\theta, \phi) = exp\left\{\frac{(y\theta – b(\theta)}{a(\phi)}) + c(y, \phi) \right\}$
does $a(\phi)$, $b(\theta)$ and $c(y, \phi)$ similarly have names? This notation is from Nelder and McCullaghs Generalized Linear Models
Best Answer
The exponential dispersion family is most easily compared to the natural exponential family (which is like the exponential family with $T(y) = y$).
The natural exponential family
$$f(y|\theta) = \exp\left(\theta^Ty-A(\theta)\right)\cdot h(y)$$
The exponential dispersion family
$$f(y|\theta,\lambda) = \exp\left(\theta^Ty-\lambda A(\theta)\right) \cdot h(y,\lambda)$$
You get the same functions $A(\theta)$ and $h(x)$, but now there is an additional dependency on a parameter $\lambda$ which scales the precision (inverse of variance) of the distribution (and also adds an additional dependency of the precision on $y$).
Your expression with $a$, $b$ and $c$ rearranges these terms, but that does not make the difference between the exponential family and the exponential dispersion family.