Link to actual problem [2534] \[ \boxed {z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (z +1\right ) y=0} \] With the expansion point for the power series method at \(z = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{2 i \sqrt {z}} \sqrt {\frac {8 z^{\frac {5}{2}}+2 z^{\frac {3}{2}}+4 i z^{2}+i z}{2 \sqrt {z}-i}}\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {{\mathrm e}^{-2 i \sqrt {z}} y}{\sqrt {-\frac {4 \left (2 \sqrt {z}+i\right ) \left (z +\frac {1}{4}\right ) z}{-2 \sqrt {z}+i}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-2 i \sqrt {z}} \sqrt {\frac {8 z^{\frac {5}{2}}+2 z^{\frac {3}{2}}-4 i z^{2}-i z}{2 \sqrt {z}+i}}\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {{\mathrm e}^{2 i \sqrt {z}} y}{\sqrt {-\frac {4 \left (-2 \sqrt {z}+i\right ) \left (z +\frac {1}{4}\right ) z}{2 \sqrt {z}+i}}}\right ] \\ \end{align*}