Renewal functions as series

Emphasis is given to accurate calculation of renewal functions for different pdfs using both convolution series with non-integral power terms and multi-segment Chebyshev polynomial series. Ten life distributions are considered: Weibull, gamma, generalised gamma, inverse Gaussian, truncated normal, extreme value I, beta-Pearson VI, Birnbaum-Saunders, lognormal, Heyde-lognormal. For the first eight, renewal function convolution series are found using Laplace transforms for single and multi component distributions in ordinary, modified or alternating renewal processes. The series have infinite regions of convergence, except for the beta-Pearson VI, but are impractical to calculate beyond small times. Also, solutions of renewal function integral equations are found using spectral collocation of Chebyshev polynomial series. Sequential solution over time segments provides flexibility and accuracy which are illustrated with figures and tabular values.