Hello dear colleagues,
I want to ask related to asymptotic behaviour of three-parameter Mittag-Leffler function type function.
It is known that two-parameter Mittag-Leffler function is going to zero, whenever argument z tends to zero for |arg(z)|>\alpha*pi/2. Is this relation is true for Prabhakar function for general values of gamma. For special cases, when gamma=1;2, it is true, because Prabhakar function can be expressed via two-parameter Mittag-Leffler function. So, can we say the same results in general case? Thanks in advance.
Best regards
Ismail T. Huseynov
Maybe this can be proved using the relationship between the two-parameter and three-parameter Mittag-Leffler functions? In the following paper we showed that the three-parameter ML function can be written as a fractional differintegral of the two-parameter ML function times a power function.
https://www.mdpi.com/2227-7390/7/6/485
Take your estimate for the two-parameter ML function, multiply it by a power function, and apply a fractional differintegral. This should give a meaningful result at least in the case where gamma
Dear Dr. Arran Fernandez, thank you for a nice idea. I will try to find asymptotic expansion of Prabhakar function via the formula which you mention in your paper. Yes you are right, from analytical continuation of RL derivative , we can consider RL integral of ML type function whenever \rho
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