  This paper deals with the series solution of a linear delay differential
equation (DDE) y'(x) = ay(x)+ by(q x), 0<q<1 with proportional delay. We
discuss the convergence of this novel series. We establish the relation between
the special function given in terms of this series and its differentials. We
also discuss the bounds on this function and present the relation with other
special functions viz. Kummer's functions, generalized Laguerre polynomials,
incomplete gamma function, beta function and regularized incomplete beta
function. Further, we discuss various properties and contiguous relations for
the novel special function. Finally, we generalize this series by solving
fractional order DDE and a system of DDE.
