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The $q$-Onsager algebra $O_q$ has a presentation involving two generators$W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Thealternating central extension $\mathcal O_q$ has a presentation involving thealternating generators $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace\mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcalG_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tildeG}_{k+1}\rbrace_{k=0}^\infty$ and a large number of relations. Let $\langle\mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal O_q$generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists analgebra isomorphism $O_q \to \langle \mathcal W_0, \mathcal W_1 \rangle$ thatsends $W_0\mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. It is knownthat the center $\mathcal Z$ of $\mathcal O_q$ is isomorphic to a polynomialalgebra in countably many variables.  It is known that the multiplication map $\langle \mathcal W_0, \mathcal W_1\rangle \otimes \mathcal Z \to \mathcal O_q$, $ w \otimes z \mapsto wz$ is anisomorphism of algebras. We call this isomorphism the standard tensor productfactorization of $\mathcal O_q$. In the study of $\mathcal O_q$ there are twonatural points of view: we can start with the alternating generators, or we canstart with the standard tensor product factorization. It is not obvious howthese two points of view are related. The goal of the paper is to describe thisrelationship. We give seven main results; the principal one is an attractivefactorization of the generating function for some algebraically independentelements that generate $\mathcal Z$.

The $q$-Onsager algebra and its alternating central extension