6.2.2.1 step 1

Assuming that the necessary conditions for case two are satisfied and \(z''=r z, r=\frac {s}{t}\). Let \(\Gamma \) be the set of all poles of \(r\). For each pole \(c\) in this set, \(E_c\) is found as follows

1. If the pole \(c\) has order \(1\) then \(E_c=\{4\}\).
2. If the pole \(c\) is of order \(2\) then \(E_c=\{2, 2+ 2\sqrt {1+ 4b}, 2-2 \sqrt {1+ 4b}\}\) where \(b\) is the coefficient of \(\frac {1}{(x-c)^2}\) in the partial fraction decomposition of \(r\). In the above set \(E_c\), only integer values are kept.
3. If the pole \(c\) is of order \(v>2\) then \(E_c=\{v\}\)

The next step is to determine \(E_{\infty }\).

1. If \(\mathcal {O}(\infty )>2\) then \(E_{\infty }=\{0,2,4\}\)
2. If \(\mathcal {O}(\infty )=2\) then \(E_{\infty }=\{2, 2+ 2\sqrt {1+ 4b}, 2-2 \sqrt {1+ 4b}\}\) where \(b=\frac {\operatorname {lcoef}(s)}{\operatorname {lcoeff}(t)}\) where \(r=\frac {s}{t}\). \(\operatorname {lcoef}(s)\) is the leading coefficient of \(s\) and similarly \(\operatorname {lcoef}(t)\) is the leading coefficient of \(t\). In the above set \(E_{\infty }\) only integer values are kept.
3. If \(\mathcal {O}(\infty )<2\) then \(E_{\infty }=\mathcal {O}(\infty )\).