Indicial Equation and Recurrence Relation

As stated earlier, the first stage is to find the values for c, through an indicial equation. This is obtained by inspecting the coefficients of the lowest powers in the respective series expansions. Consider Eq. 3.29 with P x), Qix) given by Eqs. 3.30 and 3.31 first, perform the differentiations 

This can be satisfied three possible ways, only one of which is nontrivial 

This quadratic equation is called the indicial relationship. Rearrangement shows it yields two possible values of c 

The remainder of the solution depends on the character of the values Cj, C2. If they are distinct (not equal) and do not differ by an integer, the remaining analysis is quite straightforward. However, if they are equal or differ by an integer, special techniques are required. 

We shall treat these special cases by way of example. The special cases can be categorized as  

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