  We derive a scaling property from a fundamental nonlinear differential
equation whose solution is the so-called q-exponential function. A scaling
property has been believed to be given by a power function only, but actually
more general expression for the scaling property is found to be a solution of
the above fundamental nonlinear differential equation. In fact, any power
function is obtained by restricting the domain of the q-exponential function
appropriately. As similarly as the correspondence between the exponential
function and Shannon entropy, an appropriate generalization of Shannon entropy
is expected for the scaling property. Although the q-exponential function is
often appeared in the optimal distributions of some one-parameter generalized
entropies such as Renyi entropy, only Tsallis entropy is uniquely derived from
the algebra of the q-exponential function, whose uniqueness is shown in the two
ways in this paper.
