in the Frobenius series is a root of the indicial equation
Assuming that the singular point is , we can calculate as follows:
Starting with the differential equation
Rewrite it in the form
Multiply each term by the factor .
Regroup the second and third terms as follows.
Use series for all the terms including .
Making the substitutions we get
Move the terms
the summations where they belong
Now look at the first term in each of the series, multiply and add as indicated. Mathematica can be of assistance for this computation by changing the upper limit in the summations from to .
This equation is and
we can cancel the common factor and
We have arrived at the indicial equation which was our goal. We are done!
(c) John H. Mathews 2004