Special formal series solutions of linear operator equations

Abstract

The transformation which assigns to a linear operator L the recurrence satisfied by coefficient sequences of the polynomial series in its kernel, is shown to be an isomorphism of the corresponding operator algebras. We use this fact to help factoring q-difference and recurrence operators, and to find ‘nice’ power series solutions of linear differential equations. In particular, we characterize generalized hypergeometric series that solve a linear differential equation with polynomial coefficients at an ordinary point of the equation, and show that these solutions remain hypergeometric at any other ordinary point. Therefore, to find all generalized hypergeometric series solutions, it suffices to look at a finite number of points: all the singular points, and a single, arbitrarily chosen ordinary point. We also show that at a point x= a we can have power series solutions with: • polynomial coefficient sequence — only if the equation is singular at a+1, • non-polynomial rational coefficient sequence — only if the equation is singular at a.