In Macdonald's book, the Jack symmetric function $J_{\lambda}(x_1,\ldots, x_n)$ for a partition $\lambda$ is defined by three properties (orthogonality, triangularity, and normalization). In the following paper (http://www-math.mit.edu/~rstan/pubs/pubfiles/73.pdf) its existence and uniqueness appear as Theorem 1.1. The Jack symmetric functions can be seen as eigenfunctions for the operators $$D(\alpha)= \alpha/2 \sum_{i=1}^{n}x_i^2\frac{\partial}{\partial_i^2}+\sum_{i\neq j}\frac{x_i^2}{x_i -x_j}\frac{\partial}{\partial x_i}$$ with the eigenvalues as given in Theorem 3.1 of the above paper.

Now a recent paper of Chapuy and Dolega (https://arxiv.org/pdf/2004.07824.pdf) defines the following operator, which is given in the power symmetric basis,

$$D_{\alpha}^{'}= \alpha/2 \sum_{i,j\geq 1}ij p_{i+j}\frac{\partial^2}{\partial p_i \partial p_j} + 1/2 \sum_{i,j\geq 1}(i+j) p_{i}p_{j}\frac{\partial^2}{\partial p_{i+j}}+(\alpha -1)\sum_{i\geq 1}\frac{i(i -1)}{2}p_i\frac{\partial}{\partial p_i}$$ and it defines the Jack symmetric functions to be those function which are eigenfunctions to these operators; they give the eigenvalues in terms of $\alpha$, which takes a nice form. In the paper it appears in Proposition 5.1.

My question is how to derive this operator from Stanley's paper and to apply it on the Jack symmetric function. Do we need to express the Jack symmetric function in the power symmetric basis? I cannot do it in the case of a general partition. Also, the operator in Stanley's paper involves finitely many variables, but in the Chaupy and Dolenga paper it involves infinitely many. I hope someone can give me more details.