Integro-differential Equations and Generalized Hypergeometric Functions

This paper is concerned with the integro-differential equations 1 1 y9 t. say t. qHy qt. dm q.qHy9 qt. dn q. 0 0 and 1 1 y t.qHy qt.dm q.qHy9 qt.dn q.s0, 0 0 where a is a complex constant, and m and n are complex-valued functions of bounded variation onw0, 1x. The main motivation is that the generalized hypergeo- metric function AFB a1, . . . , aA; b1, . . . , bB; t.satisfies the first equation when AFB and the second equation when AsBq1 for appropriately chosen constant a and smooth functions m and n . The first equation also includes as a special case the well-known pantograph equation and many of its generalisations. The main objects of this paper are well-posedness of initial value problems, Dirichlet and Dirichlet]Taylor series expansions, and asymptotic behaviour of the solutions