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. 



Jack function in power symmetric basis
This explanation can be found in Macdonald's book "Symmetric Functions and Hall Polynomials" by looking at Ex.VI.4.3.


Note that Stanley's Laplace-Beltrami operator $D(\alpha)$ depends on $n$ and acts on the algebra $\mathbb{Q}(\alpha)\otimes \Lambda^n$, where $\Lambda^n$ denotes the algebra of symmetric polynomials in $n$ variables $x\_1,\dots,x\_n$. Because of that I prefer to modify your notation and denote Stanley's operator by $D\_n(\alpha)$. Let me introduce a modified version of this operator $D'\_n(\alpha)$, which also acts on $\mathbb{Q}(\alpha)\otimes \Lambda^n$ and is defined by setting
$$ D'\_n(\alpha)f := \big(D\_n(\alpha)-(n-1)\deg(f)\big)f$$
for homogenous $f$ and extended by linearity.
Recall that the algebra $\Lambda$ of symmetric functions is defined as the projective limit of $\Lambda\_n$ with respect to the morphism $\rho\_n : \Lambda\_{n+1}\to \Lambda\_n$ which kills the last variable: 
$$\rho\_n(f)(x\_1,\dots,x\_n) := f(x\_1,\dots,x\_n,0).$$
It is easy to check that
$$\rho\_n D'\_{n+1}(\alpha) = D'\_{n}(\alpha)\rho\_{n},$$
so you can define an operator $D'\_\alpha := \lim D'\_{n}(\alpha)$ which acts on $\mathbb{Q}(\alpha)\otimes \Lambda$. Stanley's computations used in the proof of his Theorem 3.1 give you immediately an expression for $D'\_\alpha$ as a differential operator in power-sums. 


