On a generalization of the Bessel function Neumann expansion

The Bessel-Neumann expansion (of integer order) of a function $g:\mathbb{C}\to \mathbb{C}$ corresponds to representing $g$ as a linear combination of basis functions ${\varphi }_0,{\varphi }_1,\dots$, i.e., $g(z)={\sum }_{\mathrm{\ell }=0}^{\mathrm{\infty }}{w}_{\mathrm{\ell }}{\varphi }_{\mathrm{\ell }}(s)$, where ${\varphi }_i(z)=J_i(z)$, $i=0,\dots$, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.