Long scan duration remains a challenge for high-resolution MRI. Several accelerated imaging strategies have been proposed based on deep learning (DL) that require databases of fully-sampled images for training. However, scan-specific training is desired where individual variability is important, e.g. in free-breathing cardiac MRI, or where such datasets are not available due to scan time constraints for acquiring fully-sampled data. Building on our earlier method called Self-consistent Robust Artificial-neural-networks for k-space Interpolation (sRAKI), we propose a scan-specific DL reconstruction method based on recurrent neural networks that combines training and reconstruction phases of sRAKI. We use self-consistency among coils in k-space and regularization in arbitrary domains, as well as consistency with acquired data, in each iteration of the recurrent network. Results on knee MRI show that this method improves upon parallel imaging and compressed sensing methods.