RMSprop is an unpublished, adaptive learning rate optimization algorithm first proposed by Geoff Hinton in lecture 6 of his online class "Neural Networks for Machine Learning". RMSprop and Adadelta have been developed independently around the same time, and both try to resolve Adagrad's diminishing learning rate problem. [1]
$$E[g^2]t = 0.9 E[g^2]{t-1} + 0.1 g^2_t$$
$$\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{E[g^2]t + \epsilon}} g{t}$$
The difference between Adadelta and RMSprop is that Adadelta removes the learning rate
[1] Sebastian Ruder (2016). An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747.