  We presented background information about various entropies in the
literature. The pathway idea of Mathai (2005) is shown to be inferable from the
maximization of a certain generalized entropy measure and established
connections to outstanding problems in astronomy and physics. In this paper we
proved that the generalized entropy of Mathai associated with diffusive
processes grows linearly with the logarithm of time, and the rate of growth is
indepen- dent of the generalized entropy parameter. We also proposed some
results concerning images of generalized Bessel function under the pathway
operator and its special cases including some trigonometric functions.
Situations are listed where a generalized entropy of order ? leads to pathway
models, exponential and power law behavior and related differential equations.
