We prove that the classical normal distribution is infinitely divisible with
respect to the free additive convolution. We study the Voiculescu transform
first by giving a survey of its combinatorial implications and then
analytically, including a proof of free infinite divisibility. In fact we prove
that a subfamily Askey-Wimp-Kerov distributions are freely infinitely
divisible, of which the normal distribution is a special case.

We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and more
general expressions), where $A_N$ and $B_N$ are $N \times N$ matrices with
entries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcal
B$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haar
distributed quantum unitary random matrix with entries independent from
$\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ and
$B_N$ are asymptotically free with amalgamation over $\mathcal B$.

We provide a simple algorithm for finding the optimal upper bound for sums of
products of matrix entries of the form

S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ...
t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be
equal. The upper bound is easily obtained from a graph G associated to the
constraints in the sum.

We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.