University of Cambridge Integro-diierential Equations and Generalized Hypergeometric Functions Integro-diierential Equations and Generalized Hypergeometric Functions Are Neither Zero nor a Negative In- Teger, and 2 0; 2) There Exist Smooth Functions and Such That the Generalized Hypergeometric Function

This paper is concerned with the integro-diierential equations y 0 (t) = ay(t) + Z 1 0 y(qt) d(q) + Z 1 0 y 0 (qt) d(q); t 0 and y(t) + Z 1 0 y(qt) d(q) + Z 1 0 y 0 (qt) d(q) = 0; t 0; where a is a complex constant, while and are complex-valued functions of bounded variation on 0; 1]. The main motivation for the study of these two equations is that for every integers B + 1 A 0 and real constants e ii t) satisses the rst equation when A B and the second equation when A = B + 1. The rst equation also includes as a special case the well-known pantograph equation and many of its generalizations. The main goals of our study are well-posedness of initial value problems, Dirichlet and Dirichlet{Taylor expansions and asymptotic behaviour of the solutions.