\begin{eqnarray*} 2xu''_1 + (1+x) u'_1 + u_1 &=& 2 \sum_{n=0}^{\infty} \Big( n + \frac12 \Big) \Big( n-\frac12 \Big) a_n x^{n-\frac12} + \sum_{n=0}^{\infty} \Big( n + \frac12 \Big) a_n x^{n-\frac12} + \\ && + \sum_{n=0}^{\infty} \Big( n + \frac12 \Big) a_n x^{n+\frac12} + \sum_{n=0}^{\infty} a_n x^{n+\frac12} \\ &=& \sum_{n=1}^{\infty} \Big( n + \frac12 \Big) 2n a_n x^{n-\frac12} + \sum_{n=0}^{\infty} \Big( n + \frac32 \Big) a_n x^{n+\frac12} \\ &=& \sum_{n=0}^{\infty} \left( \Big( n + \frac32 \Big) 2(n+1) a_{n+1} + \Big( n + \frac32 \Big) 2n a_n \right) x^{n+\frac12} \\ &=& 0. \end{eqnarray*}
It is part of a Frobenius problem, that I am looking at. Can someone explain what is happening in this? For example, why becomes it $ \Big(n+\frac{1}{2} \Big)2n?$ I have tried to do $\left( \Big(n+\frac{1}{2}\Big)\Big(n-\frac{1}{2}\Big) + \Big(n+\frac{1}{2}\Big)\right)a_{n},$ but i am not getting it right.