To get Exponential GLM, you can do:
fit = glm(formula =..., family = Gamma)
But how do you get the fitted values, prediction using new data, and Pearson's residuals easily? I used to do fitted.values(fit), predict(fit, newdata, type = 'response'), and residual(fit, type = 'pearson') after fitting a GLM as the object fit. But this isn't fitting an Exponential GLM to the object fit...