Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

We derived the equation of density operator for generalized entropy and expectation value with quantum analysis when conserved quantities exist. The is simplified conventional employed. also commutation relations, $[\hat{\rho}, \hat{H}]$ \hat{Q}^{[a]}]$, are functions $\hat{\rho}$, where $\hat{H}$ Hamiltonian, $\hat{Q}^{[a]}$ quantity. operators von Neumann entropy, Tsallis R\'enyi in case value. escort average (the normalized $q$-expectation value), commutes Hamiltonian quantities. found that argument canonical ensemble simply extended to Gibbs value, even do not commute. simple extension shown average, $\hat{\rho}$ quantity $\hat{Q}^{[a]}$: \hat{H}] = [\hat{\rho}, \hat{Q}^{[a]}]=0$. These findings imply some systems.