How about

$\lambda_k = \frac{U'_k(n)}{U_k(n)}$

where the $U_k$ denote the Chebyshev polynomials of the second kind, $U_0(x)=1$, $U_1(x)=x$, and $U_k(x)=xU_{k-1}(x)-U_{k-2}(x)$ for $k\ge 2$.

In Section 10 of http://arxiv.org/abs/1210.6768 (See in particular Remark 10.4) we try to classify "Brownian motions" on $O_n^+$. The formula above follows, if you use the co-action of the free orthogonal quantum group on the free sphere to define an action of generator of "O_n^+$-BM" on the free sphere.

How about

$\lambda_k = \frac{U'_k(n)}{U_k(n)}$

where the $U_k$ denote the Chebyshev polynomials of the second kind, $U_0(x)=1$, $U_1(x)=x$, and $U_k(x)=xU_{k-1}(x)-U_{k-2}(x)$ for $k\ge 2$.

In Section 10 of http://arxiv.org/abs/1210.6768 (See in particular Remark 10.4) we try to classify "Brownian motions" on $O_n^+$. The formula above follows, if you use the co-action of the free orthogonal quantum group on the free sphere to define an action of generator of "$O_n^+$-BM" on the free sphere.