标题： 从时滞微分方程中获得的新特殊函数 显示英文标题

标题： Novel Special Function Obtained from a Delay Differential Equation

摘要： 本文讨论了线性时滞微分方程（DDE）y'(x) = ay(x)+ by(q x)，0<q<1 的级数解，该方程具有比例时滞。 我们讨论了这种新颖级数的收敛性。 我们建立了以该级数表示的特殊函数与其微分之间的关系。 我们还讨论了该函数的界限，并给出了与其他特殊函数的关系，如 Kummer 函数、广义拉盖尔多项式、不完全伽马函数、贝塔函数和正则化不完全贝塔函数。 此外，我们讨论了这种新颖特殊函数的各种性质和相邻关系。 最后，我们通过求解分数阶 DDE 和 DDE 系统来推广该级数。 显示英文摘要

摘要： 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.