Clearly, G ¯ ¯ i j ( s ; L ) in Eqs. (1)-(5) solves the corresponding continuous time random walk problem and the equivalent generalized master equation. Equations (1)-(5) enable analyzing semi-Markovian random walks in 1D chains from a wide variety of aspects. Inversion to time domain gives the Green’s function, but also moments and correlation functions can be calculated from Eqs. (1)-(5), and then inverted into time domain (for relevant quantities). The closed-form G ¯ ¯ i j ( s ; L ) also manifests its utility when numerical inversion of the generalized master equation is unstable. Moreover, using G ¯ ¯ i j ( s ; L ) in simple analytical manipulations gives, (i) the first passage time PDF, (ii)–(iii) the Green’s functions for a random walk with a special WT-PDF for the first event and for a random walk in a circular L-state 1D chain, and (iv) joint PDFs in space and time with many arguments.