i\leqslant m}\beta_{m,i}b^{m-i}ga^i,$$ where $\beta_{m,i}\in k$ with $\beta_{m,0}=1$, $\beta_{m,1}=-m$ and $\beta_{m,2}=\frac{3}{4}m(m-1)$.

\(1) The first two equalities can be proved by induction on $i$ and $j$, respectively. The third one follows from the first two equalities.

\(2) By the relations of the generators, $ab^m$ can be expressed as $ab^m = \sum\limits_{0\leqslant i\leqslant m}\alpha_{m,i}b^{m-i}a^{i+1}$ for some $\alpha_{m,i}\in k$. Then for $1\leqslant m< p-1$, by Part (1) we have $$\begin{split}
ab^{m+1}&=(\sum\limits_{0\leqslant i\leqslant m}\alpha_{m,i}b^{m-i}a^{i+1})b\\
&=\sum\limits_{0\leqslant i\leqslant m}\alpha_{m,i}b^{m-i}(a^{i+1}b)\\
&=\sum\limits_{0\leqslant i\leqslant m}\alpha_{m,i}b^{m-i}(ba^{i+1}-\frac{i+1}{2}a^{i+2})\\
&=\sum\limits_{0\leqslant i\leqslant m}\alpha_{m,i}b^{m+1-i}a^{i+1}-\sum\limits_{0\leqslant i\leqslant
m}\frac{i+1}{2}\alpha_{m,i}b^{m-i}a^{i+2}.
\end{split}$$ Hence one gets that $\alpha_{m+1,0}=\alpha_{m,0}$, $\alpha_{m+1,m+1}=-\frac{m+1}{2}\alpha_{m,m}$ and $\alpha_{m+1,i}=\alpha_{m,i}-\frac{i}{2}\alpha_{m,i-1}$ for all $1\leqslant i\leqslant m$. From the definition of $H(\lambda, \mu)$, we know that $\alpha_{1,0}=1$ and $\alpha_{1,1}=-\frac{1}{2}$. Then by induction on $m$, it is easy to check that $\alpha_{m,0}=1$, $\alpha_{m,1}=-\frac{m}{2}$ and $\alpha_{m,2}=\frac{1}{4}m(m-1)$ for all $1\leqslant m\leqslant p-1$.

\(3) It is similar to Part (2). We also have $\beta_{1, 0}=1$, $\beta_{1, 1}=-1$, $\beta_{m+1, 0}=\beta_{m, 0}$, $\beta_{m+1, m+1}=-\frac{m+2}{2}\beta_{m, m}$ and $\beta_{m+1, i}=\beta_{m, i}-\frac{i+1}{2}\beta_{m, i-1}$ for all $1\leqslant i\leqslant m<p-1$.

\[2.3\] $H(\lambda,\mu)$ is a symmetric Hopf algebra.

From $g^n=1$ and char$k=p$, it is easy to check that $g(\sum\limits_{0\leqslant i\leqslant n-1}g^i)=\sum\limits_{0\leqslant
i\leqslant n-1}g^i$ and $(1-g^p)(\sum\limits_{0\leqslant i\leqslant
n-1}ig^i)=0$. Since $\{g^ia^{p-1}b^{p-1}|0\leqslant i\leqslant
n-1\}$ are linearly independent (see [@Cib09]), $t=(\sum\limits_{0\leqslant
i\leqslant n-1}g^i)a^{p-1}b^{p-1}$ is a non-zero element of $H(\lambda,\mu)$. 