INTEGRODIFFERENTIAL EQUATIONS AND GENERALIZED HYPERGEOMETRIC-FUNCTIONS

This paper is concerned with the integro-differential equations p'(t) = ay(t) + integral(0)(1) y(qt) d mu(q) + integral(0)(1) y'(qt) d nu(q) and y(t) + integral(0)(1) y(qt) d mu(q) + integral(0)(1) y'(qt) d nu( q) = 0, where a is a complex constant, and mu and nu are complex-value d functions of bounded variation on [0, 1]. The main motivation is tha t the generalized hypergeometric function F-A(B)(alpha(1),...,alpha(A) ; beta(1),...,beta(B); t) satisfies the first equation when A less tha n or equal to B and the second equation when A = B + 1 for appropriate ly chosen constant a and smooth functions mu and nu. The first equatio n also includes as a special case the well-known pantograph equation a nd many of its generalisations. The main objects of this paper are wel l-posedness of initial value problems, Dirichlet and Dirichlet-Taylor series expansions, and asymptotic behaviour of the solutions. (C) 1997 Academic Press.