A Finitely Generated Module Representation of the Bickley-Naylor Functions

Abstract

We present a novel and non-standard derivation of the Bickley-Naylor functions $K{i}_n,n\in {\mathbb{N}}_0.$ A $3\times 3$ system of equations is developed, the solution of which yields new explicit expressions for the Bickley-Naylor functions $K{i}_2$, $K{i}_3$, and $K{i}_4$. Using a recurrence relation, expressions follow for any other order of Bickley-Naylor functions. We then characterize the infinite set of $K{i}_n,n\in {\mathbb{N}}_0$ as a four-dimensional span of modified Bessel and Struve functions over the ring of real polynomials, thus providing a new computational and modeling tool for studying and applying the Bickley-Naylor functions