Mittag-Leffer derivative

Evaluates integer-order derivatives of the Mittag-Leffler (ML) function

% ML_DERIV   Integer-order derivatives of the Mittag-Leffler function
%
%   Ek = ML_DERIV(Z, ALPHA, K) evaluates the K-th integer-order derivative 
%   of the one-parameter Mittag-Leffler function E_{ALPHA}(Z). Z may be a 
%   real or complex scalar or vector. ALPHA must be a positive real scalar. 
%   K must be a scalar integer.
%
%   Ek = ML_DERIV(Z, ALPHA, BETA, K) evaluates the K-th integer-order 
%   derivative of the two-parameter Mittag-Leffler function E_{ALPHA,BETA}(Z). 
%   Z may be a real or complex scalar or vector. ALPHA must be a positive  
%   real scalar. BETA must be a real scalar. K must be a scalar integer.
%
%   Note: Accuracy may decrease for derivatives of very high order K.
%
%   TECHNICAL NOTES
%   The derivatives are computed using an algorithm that combines a 
%   summation formula based on the Prabhakar function with a numerical 
%   Laplace-transform inversion method [2], guided by the derivatives-
%   balancing technique introduced in [1]. A full description of the 
%   algorithm and its applications is provided in [3]. If this function is
%   used for research or publications, please cite [3].
%
%   REFERENCES
%   [1] R. Garrappa and M. Popolizio, Computing the matrix Mittag-Leffler 
%       function with applications to fractional calculus, J. Sci. Comput.,
%       2018, 17(1), 129–153.
%
%   [2] R. Garrappa, Numerical Evaluation of two- and three-parameter 
%       Mittag-Leffler functions, SIAM J. Numer. Anal., 2015, 53(3), 
%       1350–1369.
%
%   [3] D. Biolek, R. Garrappa, F. Mainardi, M. Popolizio, Derivatives of 
%       Mittag-Leffler functions: theory, computation and applications, 
%       Nonlinear Dynamics, 113, 34389–34403 (2025).
%       https://doi.org/10.1007/s11071-025-11682-3
%
%  Please, report any problem or comment to : 
%          roberto dot garrappa at uniba dot it
%
%  Copyright (c) 2025
%
%  Authors:
%   Roberto Garrappa (University of Bari, Italy)
%   roberto dot garrappa at uniba dot it
%   Homepage: https://www.dm.uniba.it/members/garrappa