论文标题
$ u_q（\ wideHat {\ mathfrak {sl}} _ 2）$ u_q（\ wideHat {\ slfrak
A generating function associated with the alternating elements in the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
论文作者
论文摘要
$ u_q（\ wideHat {\ mathfrak {sl}} _ 2）$ u_q（\ wideHat {\ wideHat {\ sl}} _ 2）$的正零件$ u_q^+$ $ us $ u_q^+$ unged $ u_q^+$ $ u_q^+$ $ u_q^+$ $ u_q^+$ $ u_q^+$ $ u_q^+$ $ u_q^+$ M. Rosso于1995年引入了此嵌入。2019年，Terwilliger介绍了交替的元素$ \ {w _ { - n} _ {n \ in \ Mathbb {n}}} $，$ \ {w_ $ \ {g_ {n+1} \} _ {n \ in \ Mathbb {n}} $，$ \ {\ tilde {g} _ {n+1} _ {n+1} _ {他表明交替元素$ \ {w _ { - n} \} _ {n \ in \ mathbb {n}} $，$ \ {w_ {n+1} \} _ {n \ $ \ {\ tilde {g} _ {n+1} \} _ {n \ in \ mathbb {n}} $形成$ u_q^+$的PBW基础，他表示$ \ {g_ {g_ {n+1}}} _ {n n \ in \ in \ in \ in \ in}在他的计算中，Terwilliger使用了以下属性：使用某些元素$ \ {d_n \} _ {n \ in \ mathbb {n}} $与以下属性：生成函数$ d（t）= \ sum_ {n \ in \ sum_ { $ \ tilde {g}（t）= \ sum_ {n \ in \ mathbb {n}}} \ tilde {g} _nt^n $其中$ \ tilde {g} _0 = 1 $。 terwilliger定义$ \ {d_n \} _ {n \ in \ mathbb {n}}} $ recoursifial;在本文中，我们将以封闭的形式表达$ \ {d_n \} _ {n \ in \ mathbb {n}} $。
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.
