Special Functions

Abstract

We derive the solution of the Hermite differential equation that occurs in the ID harmonic oscillator problem. The idea of how to generate recursion relations ‘by hand’ in a symbolic computing environment for polynomials that satisfy an ordinary differential equation (ODE) can be found in the pioneering book by J. Feagin [Fe94]. There are various options to attack the problem in Maple. Sometimes one can find solutions in terms of built-in special functions, such as the Bessel functions, or the hypergeometric function, that are known to Maple. These can be found by a straight call to dsolve. More often, one is, however, unsuccessful with such a direct attempt and the expression of the solution in terms of the hypergeometric function is not necessarily very useful for a further understanding. It is possible to find solutions in series form from dsolve, which is obtained by invoking the series option for ODEs that are linear and have polynomial coefficients. The series solution may be helpful in gaining some understanding, and more importantly can be used directly to extract recursion relations for the coefficients of the series. This is explained in detail in the text by A. Heck [He93, chapter 16], and we will not use that method in this chapter, as we wish to emphasize how recursion relations are derived. For practical purposes, however, the use of the powseries package and the command for a truncated power series tpsform is generally recommended. There are help pages in Maple for the package and the commands mentioned above.