A recurrence relation for the average singular value of a complex Gaussian random matrix

In this brief note, we obtain a recurrence relation for the average singular value $\alpha \left(d\right)$ of a complex valued $d\times d$ matrix $\frac{1}{\sqrt{d}}X$ with Gaussian entries, of the form: $\begin{array}{c}{\alpha }_C\left(d,+,1\right)={\alpha }_C\left(d\right)+R\left(d\right)\text{,}\end{array}$ where R(d) can be written as the linear combination of two integrals involving Laguerre polynomials. This reduces to showing that $R\left(d\right)<0$ (a non-trivial analysis problem) a 2016 conjecture of Bandeira, Kennedy and Singer, where it is predicted that $\alpha \left(d\right)$ decreases monotonically with d, from ${\alpha }_C\left(1\right)=\sqrt{\frac{\pi }{4}}$ , to the $d\to \infty$ limit $\frac{8}{3\pi }$ provided by the Marchenko-Pastur distribution. While this may point to a possible path to solving the conjecture, it is still far from a solution, since the two integrals R(d) are non-summable hypergeometric functions, and showing that $R\left(d\right)<0$ is a non-trivial analysis problem.