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# 3 Fractional dynamics (FD) and excess wings

[123.6.1] The theory of fractional dynamics yields a three parameter function, that allows to fit both, the asymmetric peak and the excess wing with a single stretching exponent [17, 16, 23]. [123.6.2] The three parameter function is denoted as “fractional dynamics” (FD) relaxation in figure 2. [123.6.3] Its functional form reads

 (8a) (8b)

where

 (9)

with and is the binomial Mittag-Leffler function [25]. [123.6.4] The function from eq. (8) solves the fractional differential equation

 (10)

with , , and inital value [25]. [123.6.5] In eq. (10) the operator is the infinitesimal generator of fractional time evolutions of index [7, 11, 10, 9, 15, 23]. [123.6.6] It can be written as a fractional time derivative of order in the form

 (11)

where is the infinitesimal generator of translations. [123.6.7] For a mathematical definition of see [23]. [123.6.8] Note, that the solution (8) of eq. (10) holds also for generalized Riemann-Liouville operators of order and type as shown in [17, 16]. [123.6.9] The (right-/left-sided) generalized Riemann-Liouville fractional derivative of order and type with respect to was introduced in definition 3.3 in [15, p.113] by

 (12)

where

 (13)

for , denotes the right-sided Riemann-Liouville fractional integral of order , and the left sided integral is defined analogously [15, 13].