Consider a random variable that has a non-central chi-squared distribution \begin{eqnarray*} L & = & \chi_{1}^{2}(b^{2}), \end{eqnarray*} where$\chi_{1}^{2}(b^{2})$ represents a non-central chi-squared with one degree of freedom. In fact $\chi_{1}^{2}(b^{2})$ is the square of $\mathcal{N}(b,1)$. How can we write $L$ as a Gamma distribution please? I know that if $b=0,$ we can write \begin{eqnarray*} L & \sim & \Gamma(\frac{1}{2},2). \end{eqnarray*} What happens when $b\neq0$ please? Thanks.