For Fisher's method of Iteratively Reweighted Least-Squares and Newton-Raphson optimisation of the log-likelihood, the vector of parameters $\beta$ is updated using an iteration

$$\beta^{(m+1)}=\beta^{(m)}+M\left(\beta^{(m)}\right)^{-1} U\left(\beta^{(m)}\right),$$

for a specific function $M$. How is $M$ defined in each method?

Prove that they are identical in a Generalised Linear Model with the canonical link function.