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