Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ as \begin{align*} P_{\kappa}(\mathbf{z};\alpha) &=\sum_{\rho\leq\kappa} c_{\kappa,\rho}^{\alpha}\cdot m_{\rho}(\mathbf{z}) \qquad \text{and} \qquad c_{\kappa,\kappa}^{\alpha}=1 \end{align*} where the coefficients are computed recursively by \begin{align*} c_{\kappa,\rho}^{\alpha} &=\frac{2}{\alpha(u_{\kappa}^{\alpha}-u_{\rho}^{\alpha})} \sum_{\rho<\lambda\kappa} \left((r_i+s)-(r_j+s)\right)c_{\kappa,\lambda}^{\alpha} \qquad \text{while} \\ u_{\kappa}^{\alpha} &=\sum_{j=1}^n\kappa_j\left(\kappa_j-1-\frac2{\alpha}(j-1)\right) \end{align*} and $\rho=(r_1,\dots,r_n), \lambda=(r_1,\dots,r_i-s,\cdots,r_j+s,\dots,r_n)$.
Caveat: $\lambda$ must be ordered as a partition (weakly decreasing) and inequalities are in lexicographic order.
The following numerical observations are tested at least when $\alpha=\pm2$ and $\kappa=(2(n-1),2(n-2),\dots,2,0)$, which are in my immediate interest.
QUESTION. Is this true? If $u_{\kappa}^{\alpha}=u_{\rho}^{\alpha}$ for a given (specific) $\alpha$ then the coefficient $c_{\kappa,\rho}^{\alpha}=0$.
