The solution of the confluent

hypergeometric differential equation [3] 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 [34]

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 [23] by reducing the corresponding system to confluent

hypergeometric differential equations. More recently, in [24], 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.