Basic properties of incomplete Macdonald function with applications

TL;DR

This paper explores the fundamental properties of the incomplete Macdonald function, including recurrence, differential relations, and asymptotic expansions, and demonstrates its role as a solution to a relevant PDE in diffusion phenomena.

Contribution

It consolidates scattered literature on the incomplete Macdonald function and establishes its basic properties and applications in solving a key PDE.

Findings

Derived recurrence and differential relations for the incomplete Macdonald function.

Provided series and asymptotic expansions of the function.

Showed the function as a solution to a parabolic PDE relevant in diffusion processes.

Abstract

The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the basic properties of the incomplete Macdonald function, such as recurrence and differential relations, series and asymptotic expansions. This paper also shows that the incomplete Macdonald function, as a simple closed-form expression, is a particular solution to a parabolic partial differential equation, which arises naturally in a wide variety of transient and diffusion-related phenomena.