Frobenius & Special Functions · Superchallenge problem

The hypergeometric equation $z(1 - z)y'' + \left(\tfrac32 - 2z\right)y' - \tfrac14y = 0$ has indicial roots $0$ and $-\tfrac12$ at $z = 0$. Let $y(z) = \sum_{n\ge0}a_nz^n$ be the Frobenius solution for $r = 0$ with $a_0 = 1$. Compute $y\!\left(\tfrac34\right)$.