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Let $U^+_q$ denote the positive part of the quantized enveloping algebra$U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentationinvolving two generators $W_0$, $W_1$ and two relations, called the $q$-Serrerelations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting ofsome elements $\lbrace E_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty$, $\lbraceE_{n \delta+ \alpha_1} \rbrace_{n=0}^\infty$, $\lbrace E_{n \delta}\rbrace_{n=1}^\infty$. In 2019 we introduced the alternating central extension$\mathcal U^+_q$ of $U^+_q$. We defined $\mathcal U^+_q$ by generators andrelations. The generators, said to be alternating, are denoted $\lbrace\mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcalW_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$,$\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$. Let $\langle \mathcalW_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal U^+_q$ generatedby $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebraisomorphism $U^+_q\to \langle \mathcal W_0, \mathcal W_1 \rangle$ that sends$W_0 \mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. Via this isomorphismwe identify $U^+_q$ with $\langle \mathcal W_0, \mathcal W_1 \rangle$. In ourmain result, we express the Damiani PBW basis elements in terms of thealternating generators. We give the answer in terms of generating functions.

The algebra $U^+_q$ and its alternating central extension $\mathcal U^+_q$