A class of phase-like states

We investigate a class of phase-like states \ nu; epsilon > with an integer parameter nu = 0, 1.... which contains the coherent phase states \ epsilon > as the special case nu = 0. Many characteristics, such as the number sta tistics and the Susskind-Glogower phase distribution, can be calculated in a closed way and are discussed. The parameter nu can be continuously interp olated and extended to a real parameter up to arbitrary nu > -1. Problems o f the definition of phase variances are discussed. The class of states is c ompared with the so-called 'phase-optimized' states and it is found that th e case nu = 2 of them is near to these states. The properties of the consid ered class of states \ nu; epsilon > provide additional arguments to consid er the coherent phase states corresponding to nu = 0 as the genuine phase-o ptimized states. Since the Lommel polynomials play a role in the Fock-state representation of 'phase-optimized' states as is shown and to prevent misi nterpretation, it is proposed to rename these states 'Lommel states'. In th e limiting case of high mean number, they make the transition to Chebyshev states of second kind and are near to the case nu = 1 of the class of phase -like states \ nu; epsilon >. The mathematical tools connected with a gener alization of the geometric series are given in an appendix.