The solution of the confluent hypergeometric differential equation
 is often expressed as a linear combination of the Kummer functions that are defined as
(11) is a homogeneous hypergeometric differential equation
. Using a new variable x = n[R.sup.[eta]] and applying the transformation u(R) = Ry(x), the result Eq.
This is the hypergeometric differential equation
and its solution is given by 
Indeed, Dwork [16, Item 7.4] has conjectured that any globally nilpotent second order differential equation has either algebraic solutions or is gauge equivalent to a weak pullback of a Gauss hypergeometric differential equation
with rational parameters.
The discretization of the hypergeometric differential equation
on the lattice x(s) [26, 27] leads to the second order difference equation of the hypergeometric type
Equation (25) is second-order differential equation that will be reduced to hypergeometric differential equation
type; by letting coth [[omega].sub.H]r =1 - 2z, we get
He showed, in particular, that the matrix valued hypergeometric function satisfies the matrix valued hypergeometric differential equation
, and conversely that any solution of the latter is a matrix valued hypergeometric function.
A similar generalized Coulomb problem for a class of general Natanzon confluent potentials is exactly solved in  by reducing the corresponding system to confluent hypergeometric differential equations
. More recently, in , the authors succeeded to solve the eigenvalue wave equation for an electron in the field of a molecule with an electric dipole moment by expanding the solutions of a second-order Fuchsian differential equation with regular singularities in cos [theta] = 1,-1,0 in a series of Jacobi polynomials with "dipole polynomial" coefficients.