**Definition (****Indicial
Equation****).** The
parameter
in the Frobenius series is a root of the indicial equation

.

Assuming that the singular point is ,
we can calculate
as follows:

and

**Derivation.**

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
and into
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
get

.

We have arrived at the indicial equation which was our
goal. We are done!

(c) John H. Mathews 2004