Let $\Lambda$ be the algebra of symmetric functions in infinitely many variables over $\mathbb{C}$.

The $n$-th power sum symmetric function $p_n$ is defined (formally) as \begin{equation} p_n=\sum_i x_i^n\ . \end{equation} The set consisting of symmetric functions $p_\mu=p_{\mu_1}\cdots p_{\mu_t}$, for all partitions $\mu=(\mu_1, \ldots, \mu_t)$, is a basis of $\Lambda$.

For any partition $\lambda$, let us denote by $J_\lambda^\alpha$ the Jack symmetric function associated with $\alpha$. This is uniquely determined by a *triangular expansion* with respect to the monomial symmetric functions and by the condition
\begin{equation}
\langle J_\lambda^\alpha, J_\mu^\alpha\rangle_\alpha = 0 \mbox{ for } \lambda\neq \mu\ ,
\end{equation}
where $\langle \cdot , \cdot \rangle_{\alpha}$ is defined over the basis of power sums as
\begin{equation}
\langle p_\lambda, p_\mu\rangle_\alpha=\delta_{\lambda,\mu} z_\lambda \alpha^{\ell(\lambda)}\ ,
\end{equation}
where $\delta_{\lambda,\mu}=\prod_a \delta_{\lambda_a,\mu_a}$, $z_\lambda=\prod_j j^{\, m_j}\, m_j!$ ($m_j=\# \{a\in \mathbb{N}\,\vert\, \lambda_a=j\}$) and $\ell(\lambda)$ is the legth of the partition $\lambda$.

**Question:** is it possible to determine explicitly an expression of $p_1^n$, with $n\geq 1$, in terms of Jack symmetric functions $J_\lambda^\alpha$?