# moment generating function of gamma distribution through log-partition function

How to drive the moment generating function of Gamma distribution using log-partition function?

Suppose $$X\sim\Gamma(\alpha,\beta)$$, gamma distribution with parameter $$(\alpha, \beta)$$. Then $$X$$ has p.d.f $$f_{\alpha,\beta}(X) = \frac{\beta^\alpha}{\gamma(\alpha)}x^{\alpha-1}\exp\{-\beta x\}$$ In the canonical exponential family form, letting $$(\eta_1,\eta_2)=(\alpha,-\beta)$$ we have $$f_{\eta_1,\eta_2}(x)\propto\eta_1\log(-\eta_2)+(\eta_1-1)\log x-\eta_2x,$$ which means the log-partition function $$A(\eta)=\log\gamma(\eta_1+1)-(\eta_1+1)\log[-\eta_2]$$

My question is how to use the formula $$M_{X}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}}$$ to derive the momemtn generating function, which has the result given by $$M_{X}(u)=\left(1-\frac{u}{\beta}\right)^{-\alpha}$$

Particularly, since $$A$$ is a function of two parameters, i.e. $$\eta_1$$ and $$\eta_2$$, I expect the $$u$$ in $$M_{X}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}}$$ should also have two parameters, i.e. $$u_1$$ and $$u_2$$. But moment generating function of $$X$$ is a function of $$u\in\mathbb{R}$$, which is one-dimensional.

The correct formula for Moment generating function should be $$M_{T}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}},$$ not $$M_{X}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}}$$
Here, $$T(X)=(\log X,X)$$.
Thus, to get the $$M_{X}(u)$$, we need to treat $$\eta_1$$ to be a fixed constant or setting $$u_1$$ to be 0 in the first equation.