Superintegrability for two β-deformed integrals

By means of the deformed trigonometric Calogero-Sutherland (CS) operator, we construct a family of commuting operators via nested commutators. In terms of these operators, we give a hierarchy of partition functions with W -representations and present their character expansions. Moreover, certain two β -deformed integrals are constructed, where the measures are associated with the corresponding ground state wave functions of Hermite and Laguerre type CS models. We analyze the superintegrability property and show that these two integrals belong to the partition function hierarchy.