text: where  is the first solution (based on the larger root in the case of unequal roots),  is the smaller root, and the constant  and the coefficients  are to be determined. Once  is chosen (for example by setting it to 1) then  and the  are determined up to but not including , which can be set arbitrarily. This then determines the rest of the  In some cases the constant  must be zero.
  Example: consider the following differential equation (Kummer's equation with  and ):
  The roots of the indicial equation are −1 and 0. Two independent solutions are  and  so we see that the logarithm does not appear in any solution. The solution  has a power series starting with the power zero. In a power series starting with  the recurrence relation places no restriction on the coefficient for the term  which can be set arbitrarily. If it is set to zero then with this differential equation all the other coefficients will be zero and we obtain the solution .
  Tandem recurrence relations for series coefficients in the exceptional cases
  In cases in which roots of the indicial polynomial differ by an integer (including zero), the coefficients of all series involved in second linearly independent solutions can be calculated straightforwardly from tandem recurrence relations. These tandem relations can be constructed by further developing Frobenius' original invention of differentiating with respect to the parameter r, and using this approach to actually calculate the series coefficients in all cases.