given $n,N$ and show that in the limit is the “right" one as long as $n$ and $N$ tend to $\infty$.. This does not use the saddlepoint method, but relies on the availability of a generating function formula for Laguerre polynomials. The proof is done in **A7**.\
A simple modification to @imj would give the same result: in the display preceding (6.13) there, we could write $$h(t)=\sum_{k=0}^{\infty}c_k t^k = 2^{\alpha/2}\Gamma(\alpha/2)(1+t)(1-t^2)^{-(\alpha/2+1)}$$ and expand $(1-t^2)^{-(\alpha/2+1)}$. Multiplying by $1+t$ has a very simple effect on the series, and so $c_k$ is known explicitly.

In **A8**, we show how to check that the conditions required for Soshnikov’s results to hold are indeed met. They are straightforward consequences of the analysis we will carry below.

Since the real case is derived from the complex one after analyzing a few technical points, we verify these in the appendices and present here the study of the complex case. We now turn to the main problem we solve in this note: showing (\[whattoshow1\]) and (\[whattoshow2\]) under our set of assumptions.

Complex case: study of asymptotics
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In this section, we work on the problem of showing pointwise convergence and uniform boundedness, setting the problem in a way similar 