This article is a WIP - expect lots of changes in future
Kalman filtering notation is terrible. Every author uses a different, confusing, notation. I am not able to standardise it, but at least I can provide some of the key references and explain what notation they use.
Welch and Bishop
Greg Welch and Gary Bishop published the very popular An Introduction to the Kalman Filter that many people would have seen. The equations are
Predict
\[ \hat{x}^-_k = A \hat{x}_{k-1} + Bu_{k-1} \] \[ P^-_k = AP_{k-1}A^T + Q \]Update
\[ K_k = P^-_k H^T (H P^-_k H^T + R)^{-1} \] \[ \hat{x}_k = \hat{x}^-_k + K_k(z_k - H \hat{x}^-_k) \] \[ P_k = (I - K_k H)P^-_k \]Wikipedia
Wikipedia uses a more complicated notation
Predict
\[ \hat{\mathbf{x}}_{k\mid k-1} = \mathbf{F}_k \mathbf{x}_{k-1\mid k-1} + \mathbf{B}_k \mathbf{u}_{k} \] \[ \hat{\mathbf{P}}_{k\mid k-1} = \mathbf{F}_k \mathbf{P}_{k-1 \mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k \]Update
\[ \tilde{\mathbf{y}}_k = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k\mid k-1} \] \[ \mathbf{S}_k = \mathbf{H}_k \hat{\mathbf{P}}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k \] \[ \mathbf{K}_k = \hat{\mathbf{P}}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \] \[ \mathbf{x}_{k\mid k} = \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k\tilde{\mathbf{y}}_k \] \[ \mathbf{P}_{k|k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \hat{\mathbf{P}}_{k|k-1} \] \[ \tilde{\mathbf{y}}_{k\mid k} = \mathbf{z}_k - \mathbf{H}_k\mathbf{x}_{k\mid k} \]Labbe
More recently, Roger Labbe's GitHub textbook Kalman and Bayesian Filters in Python has become one of the go to texts. He uses a very simple notation
Predict
\[ \mathbf{\bar{x}} = \mathbf{Fx} + \mathbf{Bu} \] \[ \mathbf{\bar{P}} = \mathbf{FPF^T} + Q \]Update
\[ \mathbf{y} = \mathbf{z} - \mathbf{H \bar{x}} \] \[ \mathbf{S} = \mathbf{H \bar{P}} \mathbf{H}^T + R \] \[ \mathbf{K} = \mathbf{\bar{P}} \mathbf{H}^T \mathbf{S}^{-1} \] \[ \mathbf{x} = \mathbf{\bar{x}} + \mathbf{Ky} \] \[ \mathbf{P} = (\mathbf{I} - \mathbf{KH}) \mathbf{\bar{P}} \]Zarchan and Musoff
Paul Zarchan and Howard Musoff in Fundamnetals of Kalman Filtering: A Practical Approach present an unusual notation. I have mixed feelings about this textbook, since it comes with great examples of actually using the filter and an exploration of how the filter behaves. But, I hate this notation. I think it's overly complex and it doesn't make a clear, distinct separation between predictions and updates.
On the other hand, Zarchan is one of few authors who actually provides enough information to reproduce his Kalman filters. Not only that, but the methods he describes to create an extended Kalman filter seem to be the only reliable way to produce one.
\[ \hat{x}_k = \Phi_k \hat{x}_{k-1} + G_k u_{k-1} + K_k [z_k - H \Phi_k \hat{x}_{k-1} - H G_k u_{k-1}] \] \[ M_k = \Phi_k P_{k-1} \Phi^T_k + Q_k \] \[ K_k = M_k H^T [H M_k H^T + R_k]^{-1} \] \[ P_k = (I - K_k H) M_k \]Brookner
Eli Brookner in his Tracking and Kalman Filtering Made Easy presents an interesting notation
\[ X^*_{n+1,n} = \Phi X^*_{n,n} \] \[ X^*_{n,n} = X^*_{n,n-1} + H_n (Y_n - MX^*_{n,n-1}) \] \[ H_n = S^*_{n,n-1} M^T [R_n + MS^*_{n,n-1} M^T]^{-1} \] \[ S^*_{n,n-1} = \Phi S^*_{n-1,n-1} \Phi^T + Q_n \] \[ S^*_{n-1,n-1} = (I - H_{n-1} M) S^*_{n-1,n-2} \]Gelb
Arthur Gelb et al in Applied Optimal Estimation present another system for notation
\[ \hat{\underline{x}}_k(-) = \Phi_{k-1} \hat{\underline{x}}_{k-1}(+) \] \[ \hat{\underline{x}}_k(+) = \hat{\underline{x}}_k(-) + K_k[Z_k - H_k ] \hat{\underline{x}}_k(-) \] \[ K_k = P_k(-) H_k^T [H_k P_k(-) H_k^T + R_k]^{-1} \] \[ P_k(+) = \Phi_{k-1} P_{k-1}(+) \Phi^T_{k-1} + Q_{k-1} \] \[ P_k(-) = (I - K_k H_k) P_k(-) \]Brown and Hwang
Robert Grover Brown and Patrick Y.C. Hwang in Introduction to Random Signals and Applied Kalman Filtering present
\[ \hat{\mathbf{x}}^-_{k+1} = \phi_k \hat{\mathbf{x}}_k \] \[ \hat{\mathbf{x}}_k = \hat{\mathbf{x}}^-_k + \mathbf{K}_k [\mathbf{z}_k - \mathbf{H}_k \hat{\mathbf{x}}_k] \] \[ \mathbf{K}_k = \mathbf{P}^-_k \mathbf{H}^T_k [\mathbf{H}_k \mathbf{P}^-_k \mathbf{H}_k^T + \mathbf{R}_k]^{-1} \] \[ \mathbf{P}^-_{k+1} = \phi \mathbf{P}_k \phi^T_k + \mathbf{Q}_k \] \[ \mathbf{P}_k = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \mathbf{P}^-_k \]