Novel Special Function Obtained from a Delay Differential Equation

TL;DR

This paper introduces a new special function derived from the series solution of a linear delay differential equation with proportional delay, exploring its properties, relations to known functions, and generalizations to fractional and systems of DDEs.

Contribution

The paper presents a novel special function from a delay differential equation, analyzing its convergence, properties, and connections to classical functions, and extends it to fractional and system cases.

Findings

Established convergence of the series solution.

Derived relations with classical special functions.

Generalized the function to fractional and system DDEs.

Abstract

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.