Non-Hypergeometric Summation

Differential-difference equations occur in a wide variety of applications including: ship stabilization and automatic steering [82], the theory of electrical networks containing lossless transmission lines [17], the theory of biological systems [16], and in the study of distribution of primes [108]. The equation $$f'\left( t \right) + \alpha f'\left( {t - a} \right) + \beta f\left( t \right) + \gamma f\left( {t - a} \right) + \delta f\left( {t + a} \right) = 0$$ is termed a first order linear delay, or retarded, differential-difference equation for α = 0, δ = 0 and a > 0. For α = 0, δ = 0 and a 0 it is referred to as a neutral equation and when α = 0, β = 0 and a > 0, an equation of mixed type. A great deal of the studies for the stability of differential-difference equations necessitate an investigation of its associated characteristic function. Some of the early work in this area has been carried out by Pontryagin [88], Wright [114] and more recently by Cooke and van den Driessche [37] and Hao and Brauer [61]. In this chapter we will show that, by using Laplace transform techniques together with a reliance on asymptotics, series representations for the solution of differential-difference equations may be expressed in closed form. The series, in its region of convergence, it is conjectured, applies for all values of the delay parameter without necessarily relying on its association with the differential-difference equation. Unlike some of the series that are listed as high precision fraud by Borwein and Borwein [15] the series in this chapter will be shown to be exact by the use of Bürmann’s Theorem.