Hypergeometric series solutions of linear operator equations

Let K   be a field and L:K[x]→K[x]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 {bk(x)}{bk(x)} of K[x]K[x] and a hypergeometric term ckck such that y(x)=∑k=0∞ckbk(x) is a formal series solution to the equation L(y(x))=0L(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.