Hypergeometric series solutions of linear operator equations

Abstract

Let K be a field and L : K [ x ] → K [ x ] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis { b k ( x ) } of K [ x ] and a hypergeometric term c k such that y ( x ) = ∑ k = 0 ∞ c k b k ( x ) is a formal series solution to the equation L ( y ( x ) ) = 0 . This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.