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The bound state energy eigenvalues and the corresponding wave functions of the “deformed” Rosen-Morse and modified Rosen-Morse potentials have been obtained by the Nikiforov-Uvarov method (Büyükkiliç et al 1997 Theor. Chim Acta. 98 192). The “deformed hyperbolic functions” that were introduced for the first time by Arai (1991 J. Math. Anal. Appl. 158 63) have been used. The energy eigenvalues of these “deformed hyperbolic molecular potentials” bring up the criteria for the shape invariance of the potentials according to the deformation parameter q, as well as bringing a bound for the value of the parameter.

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In this study, approximate generalized quantal distribution functions and their applications, which appeared in the literature so far, have been summarized. Making use of the generalized Planck radiation law, which has been obtained by the authors of the present manuscript [Physica A240 (1997) 657], some alternative bounds for nonextensivity parameter q has been estimated. It has been shown that these results are similar to those obtained by Tsallis et al. [Phys.Rev. E52 (1995)