\[p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)=\text{(const.)}\times{\left(z^{-n}%
\phi_{2n}(z)+z^{n}\phi_{2n}(z^{-1})\right)}=\text{(const.)}\times{\left(z^{-n+%
1}\phi_{2n-1}(z)+z^{n-1}\phi_{2n-1}(z^{-1})\right)},\]