Exponential reduction of a confluent hypergeometric equation with equal parameters ($z{Y}^{\prime \prime }+(b-z)Y^{\prime }-bY=0$)

Putting $Y=e^{z}V$ in the displayed Kummer differential equation gives $z{V}^{\prime \prime }+(z+b)V^{\prime }=0$. Thus $V^{\prime }=C{z}^{-b}{e}^{-z}$ on a chosen branch, yielding $Y=e^z[A+B\int z^{-b}{e}^{-z}dz]$. For $b=1/2$ and $z=x/2$, one solution is $e^{x/2}$ and a normalized second Frobenius solution is $\sqrt{x}{\sum }_{n\ge 0}{x}^n/(2n+1)!!$. The analytic factors in both solutions have infinite radius of convergence, since their coefficient ratios tend to zero. The second solution still has a square-root branch at zero; an entire series factor does not remove a nonintegral Frobenius exponent.