The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra ${\mathcal{O}}_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra ${\mathrm{\Delta }}_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$\mathrm{.}Weapply$\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.