Confluent hypergeometric functions equation

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

This has been analytically proven [105] using the properties of the confluent hypergeometric functions i E in Equation (7.1). For an isotropic harmonic oscillator in D-dimensions, it has been shown that... 

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

To find the energy eigenvalues they imposed the boundary conditions (Equation (4)), which is equivalent to finding the zeros of the confluent hypergeometric function F at po, i.e. they must solve j F j (— a + 1, 2, po) = 0. 

One of the most utilized methods for solving the radial equation is by expanding the wave function as a power series of the radial coordinate. This approach is very appropriate because the power series [1-3,30-32,34, 35] correctly represents the confluent hypergeometric function. 

As has been known since the early studies of a spherically confined hydrogen atom [2,3]/ the required solution of Equation (10) is a confluent hypergeometric function, or Kummer M function,... 

The following equation is satisfied by the confluent hypergeometric function of Kummer... 

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