The author obtains interesting results that seem to be groundbreaking. However they are based on the identity $E_\alpha(x^\alpha)E_\alpha(y^\alpha)=E_\alpha((x+y)^\alpha)$ (Equation(2.21)), which is similar to the basic exponentiation identity $e^xe^y=e^(x+y)$. Some other authors (Guy Jumarie) also use this identity and prove it using the fractional differentiation formulas similar to the product rule and chain rule, which are generally speaking not true. On the other hand, it can be easily verified using the Matlab routine for calculating Mittag-Leffler function (http://www.mathworks.com/matlabcentral/fileexchange/8738) that the identity in question does not hold for arbitrary alpha. Can anyone provide feasible proof of the identity or otherwise clarify the issue? Thank you.
Article Theory and Applications of Local Fractional Fourier Analysis
In Yang's paper it remains unclear to me for which values of x, y, and \alpha this identity is claimed. It is clearly wrong for $\alpha = 2$ because it is well known that, in this case, $E_\alpha (x^\alpha) = E_2 (x^2) = \cosh x$, and if the identity were true in this case, then we would have $\cosh (x+y) = \cosh x \cosh y$ which contradicts the classical addition theorem for the hyperbolic cosine, $\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y$.
You might also look at Eq. (2.22) in the paper. For the special case $z_1 = 0$ and $z_2 = 1$ this reduces to $E_\alpha((-1)^\alpha) = E_\alpha((z_1 - z_2)^\alpha) = E_\alpha(z_1^\alpha) E_\alpha(-z_2^\alpha) = E_\alpha(0) E_\alpha(-1^\alpha) = E_\alpha(-1)$ (where you need to take into account that $E_\alpha(0) = 1$ for all $\alpha$). Clearly, the expression on the right-hand side is real whereas it should be quite straightforward to show that the expression on the left-hand side has a nonzero imaginary part. The same argument can be used for all $z_2 > 0$.
Therefore I believe that Yang's relation (2.21) is wrong.
Thank you for a good counterexample.
In addition, I would like to point out that the theory of Fourier-type transforms with kernels involving generalized Mittag-Leffler functions was developed in the monograph:
M.M. Dzhrbashyan: Integral Transforms and Representations of Functions in the Complex Domain, Nauka, Moscow, 1966 (in Russian).
Check this paper （A note on property of the Mittag-Leffler function）for the detailed proof (see the attachment).
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