Kummer s equation

Equation (418) may be solved in two stages first, the corresponding homogeneous equation [i.e. with the right-hand side of eqn. (418) set equal to zero] is well known as Kummer s equation and its two solutions can be written [143]... 

A second linearly independent solution of Kummer s equation, conventionally denoted U(a, b, x), is not regular at the origin and can often be excluded on physical grounds. However, as we have shown recently [2-4], when a normal hydrogen atom is confined inside a finite sphere or spherical shell, its radial eigenfunctions may involve U(a,b, x) or a combination of two linearly independent solutions of Kummer s equation, chosen so as to satisfy the boundary conditions which reflect the particular confinement considered. We have felt it necessary to address this choice in Section 2 of the present work. 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

This is Kummer s differential equation whose regular solution at the origin is the confluent hypergeometric function lFl(—a + 1, 2, p). 

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