Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 12, pp. 1-14.

Title: Analytic solutions of n-th order differential equations at a
       singular point
   
Author: Brian Haile (Northwest Missouri State Univ., Maryville, MO, USA)

Abstract:  
  Necessary and sufficient conditions are be given for the
  existence of analytic solutions of the nonhomogeneous n-th order
  differential equation at a singular point. Let $L$ be a linear
  differential operator with coefficients analytic at zero.
  If $L^*$ denotes the operator conjugate to $L$, then we will show
  that the dimension of the kernel of $L$ is equal to the dimension
  of the kernel of $L^*$.  Certain representation theorems from
  functional analysis will be used to describe the space of linear
  functionals that contain the kernel of $L^*$.  These results will
  be used to derive a form of the Fredholm Alternative that will
  establish a link between the solvability of $Ly = g$ at a singular
  point and the kernel of $L^*$. The relationship between the roots
  of the indicial equation associated with $Ly=0$ and the kernel of
  $L^*$  will allow us to show that the kernel of $L^*$ is spanned
  by a set of polynomials.  
 
Submitted July 28, 2001. Published February 4, 2002.
Math Subject Classifications: 30A99, 34A30,  34M35, 46E15.
Key Words: linear differential equation; regular
           singular point; analytic solution.