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This paper addresses a construction of new $q-$Hermite polynomials with afull characterization of their main properties and corresponding raising andlowering operator algebra. The three-term recursive relation as well as thesecond-order differential equation obeyed by these new polynomials areexplicitly derived. Relevant operator actions, including the eigenvalue problemof the deformed oscillator and the self-adjointness of the related position andmomentum operators, are investigated and analyzed. The associated coherentstates are constructed and discussed with an explicit resolution of the inducedmoment problem.

New $q-$Hermite polynomials: characterization, operator algebra and associated coherent states