Let $\lambda\in\mathbb{C}$. I want to know if the following is correct: $$\exp\left(\frac{\lambda}{2}(z+z^{-1})\right)=\exp\left(\frac{\lambda}{2}z\right)\exp\left(\frac{\lambda}{2}z^{-1}\right)=\sum_{n=-\infty}^{\infty}a_nz^n\sum_{n=-\infty}^{\infty}b_nz^n$$ with $$a_n=\begin{cases}\frac{\lambda^n}{2^nn!}&,n\ge0\\0&,n<0\end{cases}$$ $$b_n=\begin{cases}\frac{\lambda^{-n}}{2^{-n}(-n)!}&,n\le0\\0&,n>0\end{cases}.$$
Now we get $$\exp\left(\frac{\lambda}{2}(z+z^{-1})\right)=\sum_{n=-\infty}^{\infty}c_nz^n$$ with $$c_n=\sum_{k=-\infty}^{\infty}a_kb_{n-k}=\sum_{k=\max\{0,n\}}^{\infty}\frac{\lambda^k}{2^kk!}\frac{\lambda^{k-n}}{2^{k-n}(k-n)!}=\sum_{k=\max\{0,n\}}^{\infty}\frac{\lambda^{2k-n}}{2^{2k-n}(k-n)!}.$$ As I haven't used $n\ge 0$ this formula for $c_n$ is valid for every $n\in\mathbb{Z}$, not only for $n\ge0$.
Do you agree? Thanks for correcting :)