In this paper, we report research results on modeling the parameters of an improved time-frequency trajectory excitation (ITFTE) and spectral envelopes of an LPC vocoder with a long short-term memory (LSTM)-based recurrent neural network (RNN) for high-quality text-to-speech (TTS) systems. The ITFTE vocoder has been shown to significantly improve the perceptual quality of statistical parameter-based TTS systems in our prior works. However, a simple feed-forward deep neural network (DNN) with a finite window length is inadequate to capture the time evolution of the ITFTE parameters. We propose to use the LSTM to exploit the time-varying nature of both trajectories of the excitation and filter parameters, where the LSTM is implemented to use the linguistic text input and to predict both ITFTE and LPC parameters holistically. In the case of LPC parameters, we further enhance the generated spectrum by applying LP bandwidth expansion and line spectral frequency-sharpening filters. These filters are not only beneficial for reducing unstable synthesis filter conditions but also advantageous toward minimizing the muffling problem in the generated spectrum. Experimental results have shown that the proposed LSTM-RNN system with the ITFTE vocoder significantly outperforms both similarly configured band aperiodicity-based systems and our best prior DNN-trained counterpart, both objectively and subjectively.