just the elementary function $\sqrt{z}$, while the former diverges like $-e^{z}/(2z)$ for large $z$, and has thus to be discarded. Hence the solution reads g\_[-12]{}=2\^ \[eq:g-12sol\] where $\alpha^\prime$ is a constant, arbitrary at this stage.

As for the second equation in Eq. (\[eq:hierarchy\]) whose solution is the function $g_0$, it is an inhomogeneous Kummer differential equation. A basis for the solutions of the homogeneous part is, for example, the set of the two functions M(-1,12,z)=1-2z    M(-12,32,z). We need to find a particular solution of the full equation. We define $y\equiv\sqrt{z}$; The equation for $g_0$ then reads -2y+4g\_0 = 8\^( -32y\^2 +cy +34) \[eq:diffg0\] and we may look for solutions in terms of a series: g\_0(y)=\_[k=0]{}\^[+]{}a\_k y\^k . Inserting this expression into the differential equation (\[eq:diffg0\]), we get the following relations between the coefficients of the series: a\_[k+2]{}=a\_k  ,   a\_2=-2a\_0+3[\^]{} ,   a\_3=-+ ,   a\_4=-[\^]{} . The free parameters are $a_0$ and $a_1$. We may choose them in such a way that $a_{2,3}=0$: We therefore set $a_0=\frac32{\alpha^\prime}\varepsilon$ and $a_1=4{\alpha^\prime} c$. Then a\_[2n]{}=-32  ,  , where we used the duplication formula $\Gamma(2n+1)=\frac{2^{2n}}{\sqrt{\pi}}
\Gamma(n+\frac12)\Gamma(n+1)$. Switching back to the variable $z$, the final expression for the particular solution reads g\_0\^[sp]{}(z)=32[\^]{}+4[\^]{} c -F\_2(z)    F\_2(z)= \_[n=2]{}\^z\^n , 