In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
Quantum groups also apply -series extensively. Quantum groups are really not groups at all but certain Hopf algebras. They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. See Kassel (1995).
A substantial literature on -deformed quantum-mechanical Schrödinger equations has developed recently. It involves -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. See Micu and Papp (2005), where many earlier references are cited.