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

Let U+ q denote the positive part of the quantized enveloping algebra Uq(ŝl2). The algebra U+ q has a presentation involving two generators W0, W1 and two relations, called the q-Serre relations. In 1993 I. Damiani obtained a PBW basis for U+ q , consisting of some elements {Enδ+α0} ∞ n=0, {Enδ+α1} ∞ n=0, {Enδ} ∞ n=1. In 2019 we introduced the alternating central extension U+ q of U + q . We defined U + q by generators and relations. The generators, said to be alternating, are denoted {W−k} ∞ k=0, {Wk+1} ∞ k=0, {Gk+1} ∞ k=0, {G̃k+1} ∞ k=0. Let 〈W0,W1〉 denote the subalgebra of U + q generated by W0, W1. It is known that there exists an algebra isomorphism U + q → 〈W0,W1〉 that sends W0 7→ W0 and W1 7→ W1. Via this isomorphism we identify U + q with 〈W0,W1〉. 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.