# American Institute of Mathematical Sciences

September  2014, 19(7): 2267-2278. doi: 10.3934/dcdsb.2014.19.2267

## On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$

 1 Department of Physics and Astronomy, University of Bologna, and INFN, Via Irnerio 46, Bologna, I-40126, Italy

Received  April 2013 Revised  July 2013 Published  August 2014

We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_\alpha(t) := E_\alpha(-t^\alpha)$ for $0<\alpha<1$ and $t>0$, which is known to be completely monotone (CM) with a non-negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that (surprisingly) these two spectra coincide so providing a universal scaling property of this function, not well pointed out in the literature. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_\alpha(t)$ as $t\to 0$ and $t\to +\infty$. The first set is given by the stretched exponential for small times and the power law for large times, following a standard approach. For the second set we chose two rational CM functions in $t^\alpha$, obtained as the Pad\`e Approximants (PA) $[0/1]$ to the convergent series in positive powers (as $t\to 0$) and to the asymptotic series in negative powers (as $t\to \infty$), respectively. From numerical computations we are allowed to the conjecture that the second set provides upper and lower bounds to the Mittag-Leffler function.
Citation: Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267
##### References:
 [1] G. A. Baker, Essentials of Padè Approximants, Academic Press, New York, 1975.  Google Scholar [2] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific, Singapore, 2012. doi: 10.1142/9789814355216.  Google Scholar [3] L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709. doi: 10.1214/EJP.v15-762.  Google Scholar [4] E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J., Special Topics, 193 (2011) 161-171. [E-print arxiv.org/abs/1106.1761] Google Scholar [5] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134-147. [Reprinted in Fract. Calc. Appl. Anal.,10 (2007), 309-324.]  Google Scholar [6] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1 (1971), 161-198. doi: 10.1007/BF02820620.  Google Scholar [7] K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics, II. Direct current characteristics, J. Chemical Physics, 10 (1942), 98-105. doi: 10.1063/1.1723677.  Google Scholar [8] H. T. Davis, The Theory of Linear Operators, The Principia Press, Bloomington, Indiana, 1936. Google Scholar [9] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Lecture Notes in Mathematics No 2004, Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar [10] M. M. Dzherbashyan, , Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow., 1966 [in Russian]. Google Scholar [11] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar [12] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second Edition, Wiley, New York, 1971.  Google Scholar [13] A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus, {First Annual Report, NASA/TM-2002-211914}, Gleen Research Center, 2002, pp. XIV - 121. Google Scholar [14] R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), 491-518.  Google Scholar [15] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, (eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, 1997, pp. 223-276. [E-print arxiv.org/abs/0805.3823]  Google Scholar [16] B. Gross, On creep and relaxation, J. Appl. Phys., 18 (1947), 212-221. doi: 10.1063/1.1697606.  Google Scholar [17] R. Hilfer (editor), Fractional Calculus, Applications in Physics, World Scientific, Singapore, 2000. Google Scholar [18] E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Ann. Math., 31 (1930), 479-528. doi: 10.2307/1968241.  Google Scholar [19] A. A. Kilbas, A . A. Koroleva and S. V. Rogosin, Multi-parametric Mittag-Leffler functions and their extensions, Fract. Calc. Appl. Anal., 16 (2013), 378-404. doi: 10.2478/s13540-013-0024-9.  Google Scholar [20] A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type, Differential and Integral Equations, 8 (1995), 993-1011.  Google Scholar [21] A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203487372.  Google Scholar [22] A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar [23] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - New York, 1994.  Google Scholar [24] V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus, Comp. Math. Appl., 59 (2010), 1885-1895. doi: 10.1016/j.camwa.2009.08.025.  Google Scholar [25] V. Kiryakova and Y. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis, In American Institute of Physics - Conf. Proc., 1301 (2010), 597-613. doi: 10.1063/1.3526661.  Google Scholar [26] J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.  Google Scholar [27] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006. Google Scholar [28] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore, 2010. doi: 10.1142/9781848163300.  Google Scholar [29] F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.  Google Scholar [30] O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.  Google Scholar [31] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. doi: 10.1007/978-0-387-75894-7.  Google Scholar [32] A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi, 1978.  Google Scholar [33] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer Verlag, New York, 2010. doi: 10.1007/978-1-4419-0916-9.  Google Scholar [34] K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions, 12 (2001), 389-402. doi: 10.1080/10652460108819360.  Google Scholar [35] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar [36] I. Podlubny, Mittag-Leffler function, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, Matlab File Exchange www.mathworks.com/matlabcentral/fileexchange, 2006. Google Scholar [37] H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_\alpha (-x)$, Bull. Amer. Math. Soc., 54 (1948), 1115-1116. doi: 10.1090/S0002-9904-1948-09132-7.  Google Scholar [38] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993. [English translation and revised version from the Russian edition, Integrals and Derivatives of Fractional Order and Some of Their Applications Nauka i Tekhnika, Minsk, 1987]  Google Scholar [39] T. Sandev, R. Metzler and Z. Tomovski, Velocity and displacement correlation functions for fractional generalized Langevin equations, Fract. Calc. Appl. Anal., 15 (2012), 426-450. doi: 10.2478/s13540-012-0031-2.  Google Scholar [40] G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. I. Holomorphic Functions, Nordhoff, Groningen, 1960.  Google Scholar [41] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2-nd ed., De Gruyter, Berlin, 2012. doi: 10.1515/9783110269338.  Google Scholar [42] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25. [E-print arXiv:1310.1888] doi: 10.1214/EJP.v19-3058.  Google Scholar [43] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.  Google Scholar [44] A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions, Sbornik Mathematics, 198 (2007), 1011-1023. doi: 10.1070/SM2007v198n07ABEH003871.  Google Scholar [45] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2010. doi: 10.1007/978-3-642-14003-7.  Google Scholar [46] Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814. doi: 10.1080/10652461003675737.  Google Scholar [47] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013. doi: 10.1007/978-3-642-33911-0.  Google Scholar [48] R. Wong and Y.-Q Zhao, Exponential asymptotics of the Mittag-Leffler function, Constructive Approximation, 18 (2002), 355-385. doi: 10.1007/s00365-001-0019-3.  Google Scholar [49] C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse, E-print arXiv:1310.5592 [math.CA] (2013), pp. 17. Google Scholar

show all references

##### References:
 [1] G. A. Baker, Essentials of Padè Approximants, Academic Press, New York, 1975.  Google Scholar [2] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific, Singapore, 2012. doi: 10.1142/9789814355216.  Google Scholar [3] L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709. doi: 10.1214/EJP.v15-762.  Google Scholar [4] E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J., Special Topics, 193 (2011) 161-171. [E-print arxiv.org/abs/1106.1761] Google Scholar [5] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134-147. [Reprinted in Fract. Calc. Appl. Anal.,10 (2007), 309-324.]  Google Scholar [6] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1 (1971), 161-198. doi: 10.1007/BF02820620.  Google Scholar [7] K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics, II. Direct current characteristics, J. Chemical Physics, 10 (1942), 98-105. doi: 10.1063/1.1723677.  Google Scholar [8] H. T. Davis, The Theory of Linear Operators, The Principia Press, Bloomington, Indiana, 1936. Google Scholar [9] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Lecture Notes in Mathematics No 2004, Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar [10] M. M. Dzherbashyan, , Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow., 1966 [in Russian]. Google Scholar [11] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar [12] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second Edition, Wiley, New York, 1971.  Google Scholar [13] A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus, {First Annual Report, NASA/TM-2002-211914}, Gleen Research Center, 2002, pp. XIV - 121. Google Scholar [14] R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), 491-518.  Google Scholar [15] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, (eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, 1997, pp. 223-276. [E-print arxiv.org/abs/0805.3823]  Google Scholar [16] B. Gross, On creep and relaxation, J. Appl. Phys., 18 (1947), 212-221. doi: 10.1063/1.1697606.  Google Scholar [17] R. Hilfer (editor), Fractional Calculus, Applications in Physics, World Scientific, Singapore, 2000. Google Scholar [18] E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Ann. Math., 31 (1930), 479-528. doi: 10.2307/1968241.  Google Scholar [19] A. A. Kilbas, A . A. Koroleva and S. V. Rogosin, Multi-parametric Mittag-Leffler functions and their extensions, Fract. Calc. Appl. Anal., 16 (2013), 378-404. doi: 10.2478/s13540-013-0024-9.  Google Scholar [20] A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type, Differential and Integral Equations, 8 (1995), 993-1011.  Google Scholar [21] A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203487372.  Google Scholar [22] A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar [23] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - New York, 1994.  Google Scholar [24] V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus, Comp. Math. Appl., 59 (2010), 1885-1895. doi: 10.1016/j.camwa.2009.08.025.  Google Scholar [25] V. Kiryakova and Y. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis, In American Institute of Physics - Conf. Proc., 1301 (2010), 597-613. doi: 10.1063/1.3526661.  Google Scholar [26] J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.  Google Scholar [27] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006. Google Scholar [28] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore, 2010. doi: 10.1142/9781848163300.  Google Scholar [29] F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.  Google Scholar [30] O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.  Google Scholar [31] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. doi: 10.1007/978-0-387-75894-7.  Google Scholar [32] A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi, 1978.  Google Scholar [33] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer Verlag, New York, 2010. doi: 10.1007/978-1-4419-0916-9.  Google Scholar [34] K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions, 12 (2001), 389-402. doi: 10.1080/10652460108819360.  Google Scholar [35] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar [36] I. Podlubny, Mittag-Leffler function, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, Matlab File Exchange www.mathworks.com/matlabcentral/fileexchange, 2006. Google Scholar [37] H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_\alpha (-x)$, Bull. Amer. Math. Soc., 54 (1948), 1115-1116. doi: 10.1090/S0002-9904-1948-09132-7.  Google Scholar [38] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993. [English translation and revised version from the Russian edition, Integrals and Derivatives of Fractional Order and Some of Their Applications Nauka i Tekhnika, Minsk, 1987]  Google Scholar [39] T. Sandev, R. Metzler and Z. Tomovski, Velocity and displacement correlation functions for fractional generalized Langevin equations, Fract. Calc. Appl. Anal., 15 (2012), 426-450. doi: 10.2478/s13540-012-0031-2.  Google Scholar [40] G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. I. Holomorphic Functions, Nordhoff, Groningen, 1960.  Google Scholar [41] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2-nd ed., De Gruyter, Berlin, 2012. doi: 10.1515/9783110269338.  Google Scholar [42] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25. [E-print arXiv:1310.1888] doi: 10.1214/EJP.v19-3058.  Google Scholar [43] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.  Google Scholar [44] A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions, Sbornik Mathematics, 198 (2007), 1011-1023. doi: 10.1070/SM2007v198n07ABEH003871.  Google Scholar [45] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2010. doi: 10.1007/978-3-642-14003-7.  Google Scholar [46] Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814. doi: 10.1080/10652461003675737.  Google Scholar [47] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013. doi: 10.1007/978-3-642-33911-0.  Google Scholar [48] R. Wong and Y.-Q Zhao, Exponential asymptotics of the Mittag-Leffler function, Constructive Approximation, 18 (2002), 355-385. doi: 10.1007/s00365-001-0019-3.  Google Scholar [49] C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse, E-print arXiv:1310.5592 [math.CA] (2013), pp. 17. Google Scholar
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