LSTM

input gate

$$i^{(t)} = sigmoid(W^i [h^{(t-1)}, x^{(t)}] + b^i)$$

forget gate

$$f^{(t)}=sigmoid(W^f [h^{(t-1)}, x^{(t)}] + b^f)$$

output gate

$$o^{(t)}=sigmoid(W^o [h^{(t-1)}, x^{(t)}] + b^o)$$

$$\overline{C}^{(t)}=tanh(W^C[h^{(t-1)}, x^{(t)}] + b^C)$$

$$C^{(t)} = f^{(t)}C^{(t-1)} + i^{(t)}\overline{C}^{(t)}$$

$$h^{(t)} = tanh(C^{(t)})\times o^{(t)}$$