The purpose of the book is to study linear lattice vibrations by means of the recurrence relations method focusing on momentum autocorrelation function (ACF) of a tagged oscillator in different chains. A monatomic chain is composed of same kind of harmonic oscillators. Its momentum ACF is given by a zero-order Bessel function. A diatomic chain is composed of two kinds oscillators located alternatively in a circle chain. The momentum ACF is a sum of acoustic and optical branches given as even-order Bessel function expansions. The expansion coefficients are integrals of real and complex elliptic functions. The momentum ACF of a mass impurity in a monatomic chain has a cosine in addition to the cut contribution. The momentum ACF of a mass impurity in a diatomic chain results from two pairs of poles and three branch cuts. In addition, Fibonacci chains, the ergodic behavior of different chains are analysed. Finally, some relevant models are briefly discussed such as Bethe lattice, the independent oscillator model, 2-dimensional electron gas, spin dynamics and nonlinear interaction, etc.