A generating function associated with the alternating elements in the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

Abstract: The positive part $U_q^+$ of $U_q(\widehat{\mathfrak{sl}}2)$ admits an embedding into a $q$-shuffle algebra. This embedding was introduced by M. Rosso in 1995. In 2019, Terwilliger introduced the alternating elements ${W{-n}}{n \in \mathbb{N}}$, ${W{n+1}}{n \in \mathbb{N}}$, ${G{n+1}}{n \in \mathbb{N}}$, ${\tilde{G}{n+1}}{n \in \mathbb{N}}$ in $U_q^+$ using the Rosso embedding. He showed that the alternating elements ${W{-n}}{n \in \mathbb{N}}$, ${W{n+1}}{n \in \mathbb{N}}$, ${\tilde{G}{n+1}}{n \in \mathbb{N}}$ form a PBW basis for $U_q^+$, and he expressed ${G{n+1}}{n \in \mathbb{N}}$ in this alternating PBW basis. In his calculation, Terwilliger used some elements ${D_n}{n \in \mathbb{N}}$ with the following property: the generating function $D(t)=\sum_{n \in \mathbb{N}}D_nt^n$ is the multiplicative inverse of the generating function $\tilde{G}(t)=\sum_{n \in \mathbb{N}}\tilde{G}nt^n$ where $\tilde{G}_0=1$. Terwilliger defined ${D_n}{n \in \mathbb{N}}$ recursively; in this paper, we will express ${D_n}_{n \in \mathbb{N}}$ in closed form.