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## GENERALIZED BESSEL FUNCTIONS OF E. M. WRIGHT FROM EARLY 1930s

The integral representation of the Wright function is (see Gorenflo et. al.)

$\phi((\rho,\beta; z) = \frac{1}{2\pi i}\int_{Negative Real Line} e^{\zeta+z \zeta^{-\rho}}\zeta^{-\beta} d\zeta$

They show the approximate formula:

$\phi(\rho,\beta;-x) = x^{p(1/2-\beta)} e^{\sigma x^p \cos(\pi p)}\cos( Q ) ( c_1 + O(x^{-p}))$

where $p=1/(1+\rho)$ and $\sigma=(1+\rho)\rho^{-\frac{1}{1+\rho}}$.  This is a deviation from power law for the fundamental solution of the fractional diffusion equation. and we wish to discover if this exact deviation from the power law can explain the observed deviation from the power law in empirical data.