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2014 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin

Erschienen in: Mittag-Leffler Functions, Related Topics and Applications

Verlag: Springer Berlin Heidelberg

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Abstract

The book is devoted to an extended description of the properties of the Mittag-Leffler function, its numerous generalizations and their applications in different areas of modern science.

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Zurück zum Zitat Bromwich, T.J.I.: An introduction to the theory of infinite series. AMS Chelsea Publishing, Providence, RI (1926)MATH Bromwich, T.J.I.: An introduction to the theory of infinite series. AMS Chelsea Publishing, Providence, RI (1926)MATH
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Zurück zum Zitat Buckwar, E., Luchko, Yu.: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227, 81–97 (1998)MathSciNetMATH Buckwar, E., Luchko, Yu.: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227, 81–97 (1998)MathSciNetMATH
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Zurück zum Zitat Buhl, A.: Sommabilité et fonction E α (x). Enseignment. 24, 69–76 (1925)MATH Buhl, A.: Sommabilité et fonction E α (x). Enseignment. 24, 69–76 (1925)MATH
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Zurück zum Zitat Buhl, A.: Séries Analytiques. Sommabilité. Mémorial des Sciences Mathématiques. Acad. Sci. Paris, Fasc., vol. VII. Gauthier-Villars, Paris (1925) Buhl, A.: Séries Analytiques. Sommabilité. Mémorial des Sciences Mathématiques. Acad. Sci. Paris, Fasc., vol. VII. Gauthier-Villars, Paris (1925)
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Zurück zum Zitat Buschman, R.G.: Contiguous relations and related formulas for the H-function of Fox. Jñānābha Sect. A 2, 39–47 (1972)MathSciNetMATH Buschman, R.G.: Contiguous relations and related formulas for the H-function of Fox. Jñānābha Sect. A 2, 39–47 (1972)MathSciNetMATH
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Zurück zum Zitat Butzer, P., Westphal, U.: Introduction to fractional calculus. In: Hilfer, R. (ed.) Fractional Calculus, Applications in Physics, pp. 1–85. World Scientific, Singapore (2000) Butzer, P., Westphal, U.: Introduction to fractional calculus. In: Hilfer, R. (ed.) Fractional Calculus, Applications in Physics, pp. 1–85. World Scientific, Singapore (2000)
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Zurück zum Zitat Campos, L.M.: On the solution of some simple fractional differential equations. Int. J. Math. Math. Sci. 13 (3), 481–496 (1990)MathSciNetMATH Campos, L.M.: On the solution of some simple fractional differential equations. Int. J. Math. Math. Sci. 13 (3), 481–496 (1990)MathSciNetMATH
[Cap13]
Zurück zum Zitat Capelas de Oliveira, E. Capelas’s relations. Personal communication (2013) Capelas de Oliveira, E. Capelas’s relations. Personal communication (2013)
[CapMai11]
Zurück zum Zitat Capelas de Oliveira, E., Mainardi, F., Vaz, J. Jr.: Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J.-Special Topics 193, 161–171 (2011). [Revised Version as E-print http://arxiv.org/abs/1106.1761v2] Capelas de Oliveira, E., Mainardi, F., Vaz, J. Jr.: Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J.-Special Topics 193, 161–171 (2011). [Revised Version as E-print http://​arxiv.​org/​abs/​1106.​1761v2]
[Cap-et-al10]
Zurück zum Zitat Caponetto, R., Dongola, G., Fortuna, L., and Petráš, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore (2010) Caponetto, R., Dongola, G., Fortuna, L., and Petráš, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore (2010)
[Cap66]
Zurück zum Zitat Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Annali di Geofisica 19, 383–393 (1966) Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Annali di Geofisica 19, 383–393 (1966)
[Cap67]
Zurück zum Zitat Caputo, M.: Linear models of dissipation whose Q is almost frequency independent: part II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967) Caputo, M.: Linear models of dissipation whose Q is almost frequency independent: part II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)
[Cap69]
Zurück zum Zitat Caputo, M.: Elasticità e Dissipazione. Zanichelli, Bologna (1969) Caputo, M.: Elasticità e Dissipazione. Zanichelli, Bologna (1969)
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Zurück zum Zitat Caputo, M.: Vibrations of an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56(3), 897–904 (1974)MATH Caputo, M.: Vibrations of an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56(3), 897–904 (1974)MATH
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Zurück zum Zitat Caputo, M.: Which is the correct stress strain relation for the anelasticity of the earth’s interior? Geophys. J. R. Astron. Soc. 58, 432–435 (1979) Caputo, M.: Which is the correct stress strain relation for the anelasticity of the earth’s interior? Geophys. J. R. Astron. Soc. 58, 432–435 (1979)
[Cap85]
Zurück zum Zitat Caputo, M.: Generalized rheology and geophysical consequences. Tectonophysics 116, 163–172 (1985) Caputo, M.: Generalized rheology and geophysical consequences. Tectonophysics 116, 163–172 (1985)
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Zurück zum Zitat Caputo, M.: The rheology of an anelastic medium studied by means of the observation of the splitting of its eigenfrequencies. J. Acoust. Soc. Am. 86(5), 1984–1989 (1989) Caputo, M.: The rheology of an anelastic medium studied by means of the observation of the splitting of its eigenfrequencies. J. Acoust. Soc. Am. 86(5), 1984–1989 (1989)
[Cap96]
Zurück zum Zitat Caputo, M.: Modern rheology and electric induction: multivalued index of refraction, splitting of eigenvalues and fatigues. Ann. Geofis. 39(5), 941–966 (1996) Caputo, M.: Modern rheology and electric induction: multivalued index of refraction, splitting of eigenvalues and fatigues. Ann. Geofis. 39(5), 941–966 (1996)
[Cap00]
Zurück zum Zitat Caputo, M.: Models of flux in porous media with memory. Water Resourc. Res. 36(3), 693–705 (2000) Caputo, M.: Models of flux in porous media with memory. Water Resourc. Res. 36(3), 693–705 (2000)
[Cap01]
Zurück zum Zitat Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001)MathSciNetMATH Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001)MathSciNetMATH
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Zurück zum Zitat Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II). 1, 161–198 (1971) Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II). 1, 161–198 (1971)
[CapMai71b]
Zurück zum Zitat Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. (PAGEOPH) 91, 134–147 (1971). [Reprinted in Fract. Calc. Appl. Anal. 10(3), 309–324 (2007)] Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. (PAGEOPH) 91, 134–147 (1971). [Reprinted in Fract. Calc. Appl. Anal. 10(3), 309–324 (2007)]
[CarMai97]
Zurück zum Zitat Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. CISM Courses and Lecture Notes, vol. 378. Springer, Wien/New York (1997) Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. CISM Courses and Lecture Notes, vol. 378. Springer, Wien/New York (1997)
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Zurück zum Zitat Cole, K.S.: Electrical conductance of biological systems, electrical excitation in nerves. In: Proceedings Symposium on Quantitative Biology, Cold Spring Harbor Laboratory, July 1933, vol. 1, pp. 107–116. Cold Spring Harbor, Laboratory Press, New York (1933) Cole, K.S.: Electrical conductance of biological systems, electrical excitation in nerves. In: Proceedings Symposium on Quantitative Biology, Cold Spring Harbor Laboratory, July 1933, vol. 1, pp. 107–116. Cold Spring Harbor, Laboratory Press, New York (1933)
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Zurück zum Zitat Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341–351 (1941) Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341–351 (1941)
[ColCol42]
Zurück zum Zitat Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics. II. Direct current characteristics. J. Chem. Phys. 10, 98–105 (1942) Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics. II. Direct current characteristics. J. Chem. Phys. 10, 98–105 (1942)
[Cox67]
Zurück zum Zitat Cox, D.R.: Renewal Theory, 2nd edn. Methuen, London (1967)MATH Cox, D.R.: Renewal Theory, 2nd edn. Methuen, London (1967)MATH
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Zurück zum Zitat Craig, J.D., Brown, J.C.: Inverse Problems in Astronomy. Adam Hilger Ltd, Bristol (1986)MATH Craig, J.D., Brown, J.C.: Inverse Problems in Astronomy. Adam Hilger Ltd, Bristol (1986)MATH
[Dav36]
Zurück zum Zitat Davis, H.T.: The Theory of Linear Operators. The Principia Press, Bloomington (1936) Davis, H.T.: The Theory of Linear Operators. The Principia Press, Bloomington (1936)
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Zurück zum Zitat Davis, P.J.: Leonard Euler’s integral: a historical profile of the Gamma function. Am. Math. Mon. 66, 849–869 (1959)MATH Davis, P.J.: Leonard Euler’s integral: a historical profile of the Gamma function. Am. Math. Mon. 66, 849–869 (1959)MATH
[Dav02]
Zurück zum Zitat Davies, B.J.: Integral Transforms and Their Applications, 3rd edn. Springer, Berlin/New York (2002)MATH Davies, B.J.: Integral Transforms and Their Applications, 3rd edn. Springer, Berlin/New York (2002)MATH
[DebBha07]
Zurück zum Zitat Debnath, L., Bhatta, D.: Integral Transforms and Their Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007) Debnath, L., Bhatta, D.: Integral Transforms and Their Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)
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[Die10]
Zurück zum Zitat Diethelm, K.: The Analysis of Differential Equations of Fractional Order: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin/Heidelberg (2010) Diethelm, K.: The Analysis of Differential Equations of Fractional Order: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin/Heidelberg (2010)
[Die-et-al05]
Zurück zum Zitat Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)MathSciNetMATH Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)MathSciNetMATH
[DitPru65]
Zurück zum Zitat Ditkin, V.A., Prudnikov, A.P.: Integral Transforms and Operational Calculus. Pergamon Press, Oxford (1965)MATH Ditkin, V.A., Prudnikov, A.P.: Integral Transforms and Operational Calculus. Pergamon Press, Oxford (1965)MATH
[Doe74]
Zurück zum Zitat Doetsch, G.: Introduction to the Theory and Applications of the Laplace Transformation. Springer, New York (1974) Doetsch, G.: Introduction to the Theory and Applications of the Laplace Transformation. Springer, New York (1974)
[Dzh54a]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: On integral representation of functions continuous on given rays (generalization of the Fourier integrals). Izvestija Akad. Nauk SSSR Ser. Mat. 18, 427–448 (1954, in Russian) Dzherbashian [ = Djrbashian], M.M.: On integral representation of functions continuous on given rays (generalization of the Fourier integrals). Izvestija Akad. Nauk SSSR Ser. Mat. 18, 427–448 (1954, in Russian)
[Dzh54b]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: On the asymptotic expansion of a function of Mittag-Leffler type, Akad. Nauk Armjan. SSR Doklady. 19, 65–72 (1954, in Russian) Dzherbashian [ = Djrbashian], M.M.: On the asymptotic expansion of a function of Mittag-Leffler type, Akad. Nauk Armjan. SSR Doklady. 19, 65–72 (1954, in Russian)
[Dzh54c]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: On Abelian summation of the eneralized integral transform. Akad. Nauk Armjan. SSR Izvestija, fiz-mat. estest. techn. nauki. 7(6), 1–26 (1954, in Russian) Dzherbashian [ = Djrbashian], M.M.: On Abelian summation of the eneralized integral transform. Akad. Nauk Armjan. SSR Izvestija, fiz-mat. estest. techn. nauki. 7(6), 1–26 (1954, in Russian)
[Dzh60]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: On integral transforms generated by the generalized Mittag-Leffler function. Izv. Akad. Nauk Armjan. SSR 13(3), 21–63 (1960, in Russian) Dzherbashian [ = Djrbashian], M.M.: On integral transforms generated by the generalized Mittag-Leffler function. Izv. Akad. Nauk Armjan. SSR 13(3), 21–63 (1960, in Russian)
[Dzh66]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: Integral Transforms and Representation of Functions in the Complex Domain. Nauka, Moscow (1966, in Russian) Dzherbashian [ = Djrbashian], M.M.: Integral Transforms and Representation of Functions in the Complex Domain. Nauka, Moscow (1966, in Russian)
[Dzh70]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: A boundary value problem for a Sturm–Liouville type differential operator of fractional order. Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 5(2), 71–96 (1970, in Russian) Dzherbashian [ = Djrbashian], M.M.: A boundary value problem for a Sturm–Liouville type differential operator of fractional order. Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 5(2), 71–96 (1970, in Russian)
[Dzh84]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M.: Interpolation and spectral expansions associated with differential operators of fractional order. Sov. J. Contemp. Math. Anal. Arm. Acad. Sci. 19(2), 116 (1984) (English. Russian original); translation from Izv. Akad. Nauk Arm. SSR, Mat. 19(2), 81–181 (1984) Dzherbashian [ = Djrbashian], M.M.: Interpolation and spectral expansions associated with differential operators of fractional order. Sov. J. Contemp. Math. Anal. Arm. Acad. Sci. 19(2), 116 (1984) (English. Russian original); translation from Izv. Akad. Nauk Arm. SSR, Mat. 19(2), 81–181 (1984)
[Djr93]
Zurück zum Zitat Djrbashian [ = Dzherbashian], M.M.: Harmonic Analysis and Boundary Value Problems in the Complex Domain. Birkhäuser Verlag, Basel (1993) Djrbashian [ = Dzherbashian], M.M.: Harmonic Analysis and Boundary Value Problems in the Complex Domain. Birkhäuser Verlag, Basel (1993)
[DzhNer68]
Zurück zum Zitat Dzherbashian [ = Djrbashian], M.M., Nersesian, A.B.: Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Arm. SSR, Mat. 3(1), 3–29 (1968, in Russian) Dzherbashian [ = Djrbashian], M.M., Nersesian, A.B.: Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Arm. SSR, Mat. 3(1), 3–29 (1968, in Russian)
[EmKlMi01]
Zurück zum Zitat Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extreme Events for Insurance and Finance. Springer, Berlin (2001) Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extreme Events for Insurance and Finance. Springer, Berlin (2001)
[Bat-1]
Zurück zum Zitat Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953) Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)
[Bat-2]
Zurück zum Zitat Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw Hill, New York (1954) Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw Hill, New York (1954)
[Bat-3]
Zurück zum Zitat Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1955)MATH Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1955)MATH
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Zurück zum Zitat Evgrafov, M.A., et al.: Collection of Exercises on the Theory of Analytic Functions. Nauka, Moscow (1969, in Russian) Evgrafov, M.A., et al.: Collection of Exercises on the Theory of Analytic Functions. Nauka, Moscow (1969, in Russian)
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Zurück zum Zitat Evgrafov, M.A.: Asymptotic Estimates and Entire Functions, 2nd edn. Nauka, Moscow (1978, in Russian) Evgrafov, M.A.: Asymptotic Estimates and Entire Functions, 2nd edn. Nauka, Moscow (1978, in Russian)
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Zurück zum Zitat Feller, W.: Fluctuation theory of recurrent events. Trans. Am. Math. Soc. 67, 98–119 (1949)MathSciNetMATH Feller, W.: Fluctuation theory of recurrent events. Trans. Am. Math. Soc. 67, 98–119 (1949)MathSciNetMATH
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Zurück zum Zitat Feller, W.: On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them. Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, pp. 73–81 (1952) Feller, W.: On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them. Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, pp. 73–81 (1952)
[Fel71]
Zurück zum Zitat Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971) Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971)
[Fog94]
Zurück zum Zitat Fogedby, H.C.: Langevin equations for continuous time Lévy flights. Phys. Rev. E 50, 1657–1660 (1994) Fogedby, H.C.: Langevin equations for continuous time Lévy flights. Phys. Rev. E 50, 1657–1660 (1994)
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Zurück zum Zitat Fox, C.: The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. Ser. 2 27, 389–400 (1928) Fox, C.: The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. Ser. 2 27, 389–400 (1928)
[Fox61]
Zurück zum Zitat Fox, C.: The G and H functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. 98, 395–429 (1961)MATH Fox, C.: The G and H functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. 98, 395–429 (1961)MATH
[FrDiLu02]
Zurück zum Zitat Freed, A., Diethelm, K., Luchko, Y.: Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus. First Annual Report, NASA/TM-2002-211914, Gleen Research Center, pp. I–XIV, 1–121 (2002) Freed, A., Diethelm, K., Luchko, Y.: Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus. First Annual Report, NASA/TM-2002-211914, Gleen Research Center, pp. I–XIV, 1–121 (2002)
[Fuj33]
Zurück zum Zitat Fujiwara, M.: On the integration and differentiation of an arbitrary order. Tohoku Math. J. 37, 110–121 (1933) Fujiwara, M.: On the integration and differentiation of an arbitrary order. Tohoku Math. J. 37, 110–121 (1933)
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Zurück zum Zitat Garrappa, R., Popolizio, M.: Evaluation of generalized Mittag-Leffler functions on the real line. Adv. Comput. Math. 39(1), 205–225 (2013)MathSciNetMATH Garrappa, R., Popolizio, M.: Evaluation of generalized Mittag-Leffler functions on the real line. Adv. Comput. Math. 39(1), 205–225 (2013)MathSciNetMATH
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Zurück zum Zitat Gauss, C.F.: Werke, Bd. III. Analysis. (various texts, in Latin and German, orig. publ. between 1799–1851, or found in the “Nachlass”; annotated by E.J. Schering). Dieterichschen Universitäts-Druckerei W. Fr. Kaestner (1866) Gauss, C.F.: Werke, Bd. III. Analysis. (various texts, in Latin and German, orig. publ. between 1799–1851, or found in the “Nachlass”; annotated by E.J. Schering). Dieterichschen Universitäts-Druckerei W. Fr. Kaestner (1866)
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Zurück zum Zitat Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic, New York (1964) (English translation from the Russian, Nauka, Moscow, 1959)MATH Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic, New York (1964) (English translation from the Russian, Nauka, Moscow, 1959)MATH
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Zurück zum Zitat Gemant, A.: A method of analyzing experimental results obtained from elastiviscous bodies. Physics 7, 311–317 (1936) Gemant, A.: A method of analyzing experimental results obtained from elastiviscous bodies. Physics 7, 311–317 (1936)
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Zurück zum Zitat Gemant, A.: On fractional differentials. Philos. Mag. (Ser. 7) 25, 540–549 (1938) Gemant, A.: On fractional differentials. Philos. Mag. (Ser. 7) 25, 540–549 (1938)
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Zurück zum Zitat Gemant, A.: Frictional phenomena: VIII. J. Appl. Phys. 13, 210–221 (1942) Gemant, A.: Frictional phenomena: VIII. J. Appl. Phys. 13, 210–221 (1942)
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Zurück zum Zitat Gemant, A.: Frictional Phenomena. Chemical Publ. Co, Brooklyn (1950) Gemant, A.: Frictional Phenomena. Chemical Publ. Co, Brooklyn (1950)
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Zurück zum Zitat Gerasimov, A.N.: A generalization of linear laws of deformation and its application to internal friction problem. Akad. Nauk SSSR. Prikl. Mat. Mekh. 12, 251–260 (1948, in Russian) Gerasimov, A.N.: A generalization of linear laws of deformation and its application to internal friction problem. Akad. Nauk SSSR. Prikl. Mat. Mekh. 12, 251–260 (1948, in Russian)
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Zurück zum Zitat Glöckle, W.G., Nonnenmacher, T.F.: Fox function representation of non-Debye relaxation processes. J. Stat. Phys. 71, 741–757 (1993)MATH Glöckle, W.G., Nonnenmacher, T.F.: Fox function representation of non-Debye relaxation processes. J. Stat. Phys. 71, 741–757 (1993)MATH
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Zurück zum Zitat Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independently Random Variables. Addison-Wesley, Cambridge (1954) (English translation from the Russian edition, G.I.T.T.L., Moscow, 1949) Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independently Random Variables. Addison-Wesley, Cambridge (1954) (English translation from the Russian edition, G.I.T.T.L., Moscow, 1949)
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Zurück zum Zitat Gnedenko, B.V., Kovalenko, I.N.: Introduction to Queueing Theory. Israel Program for Scientific Translations, Jerusalem (1968) Gnedenko, B.V., Kovalenko, I.N.: Introduction to Queueing Theory. Israel Program for Scientific Translations, Jerusalem (1968)
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Zurück zum Zitat Gol’dberg, A.A., Levin, B.Ya., Ostrovskij, I.V.: Entire and meromorphic functions. Itogi Nauki Tekh. Ser. Sovrem. Probl. Math. Fundam. Napravleniya 85, 5–186 (1991, in Russian) Gol’dberg, A.A., Levin, B.Ya., Ostrovskij, I.V.: Entire and meromorphic functions. Itogi Nauki Tekh. Ser. Sovrem. Probl. Math. Fundam. Napravleniya 85, 5–186 (1991, in Russian)
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Zurück zum Zitat Gonsovski, V.L., Rossikhin, Yu.A.: Stress waves in a viscoelastic medium with a singular hereditary kernel. J. Appl. Mech. Tech. Phys. 14(4), 595–597 (1973) Gonsovski, V.L., Rossikhin, Yu.A.: Stress waves in a viscoelastic medium with a singular hereditary kernel. J. Appl. Mech. Tech. Phys. 14(4), 595–597 (1973)
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Zurück zum Zitat Gorenflo, R.: Abel Integral Equations with Special Emphasis on Applications. Lectures in Mathematical Sciences, vol. 13. The University of Tokyo, Graduate School of Mathematical Sciences, Tokyo (1996) Gorenflo, R.: Abel Integral Equations with Special Emphasis on Applications. Lectures in Mathematical Sciences, vol. 13. The University of Tokyo, Graduate School of Mathematical Sciences, Tokyo (1996)
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Zurück zum Zitat Gorenflo, R.: Fractional calculus: some numerical methods. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 277–290. Springer, Wien (1997) Gorenflo, R.: Fractional calculus: some numerical methods. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 277–290. Springer, Wien (1997)
[Gor98]
Zurück zum Zitat Gorenflo, R.: The tomato salad problem in spherical stereology. In: Rusev, P., Dimovski, I., Kiryakova, V. (eds.) Transform Methods and Special Functions, Varna 1996, pp. 132–149. Science Culture Technology, Singapore (1998) Gorenflo, R.: The tomato salad problem in spherical stereology. In: Rusev, P., Dimovski, I., Kiryakova, V. (eds.) Transform Methods and Special Functions, Varna 1996, pp. 132–149. Science Culture Technology, Singapore (1998)
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Zurück zum Zitat Gorenflo, R., Mainardi, F.: Fractional oscillations and Mittag-LefHer functions. In: (ed.) International Workshop on the Recent Advances in Applied Mathematics, RAAM’96, University Kuwait, D.M.C.S., Kuwait, pp. 193–208 (1996) Gorenflo, R., Mainardi, F.: Fractional oscillations and Mittag-LefHer functions. In: (ed.) International Workshop on the Recent Advances in Applied Mathematics, RAAM’96, University Kuwait, D.M.C.S., Kuwait, pp. 193–208 (1996)
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Zurück zum Zitat Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997) Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)
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Zurück zum Zitat Kilbas, A., Saxena, R.K., Saigo, M., Trujillo, J.J.: Series representations and asymptotic expansions of extended generalized hypergeometric function. In: Rogosin, S.V. (ed.) Analytic Methods of Analysis and Differential Equations: AMADE-2009, pp. 31–59. Cambridge Scientific Publishers, Cottenham (2012) Kilbas, A., Saxena, R.K., Saigo, M., Trujillo, J.J.: Series representations and asymptotic expansions of extended generalized hypergeometric function. In: Rogosin, S.V. (ed.) Analytic Methods of Analysis and Differential Equations: AMADE-2009, pp. 31–59. Cambridge Scientific Publishers, Cottenham (2012)
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Zurück zum Zitat Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
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Zurück zum Zitat Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems – I. Appl. Anal. 78(1–2), 153–192 (2001)MathSciNetMATH Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems – I. Appl. Anal. 78(1–2), 153–192 (2001)MathSciNetMATH
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Zurück zum Zitat Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems – II. Appl. Anal. 81(2), 435–494 (2002)MathSciNetMATH Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems – II. Appl. Anal. 81(2), 435–494 (2002)MathSciNetMATH
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Zurück zum Zitat Kilbas, A.A., Zhukovskaya, N.V.: Solution of Euler-type non-homogeneous differential equations with three fractional derivatives. In: Kilbas, A.A., Rogosin, S.V. (eds.) Analytic Methods of Analysis and Differential Equations: AMADE-2006, pp. 111–137. Cambridge Scientific Publisher, Cottenham/Cambridge (2008) Kilbas, A.A., Zhukovskaya, N.V.: Solution of Euler-type non-homogeneous differential equations with three fractional derivatives. In: Kilbas, A.A., Rogosin, S.V. (eds.) Analytic Methods of Analysis and Differential Equations: AMADE-2006, pp. 111–137. Cambridge Scientific Publisher, Cottenham/Cambridge (2008)
[KilZhu09a]
Zurück zum Zitat Kilbas, A.A., Zhukovskaya, N.V.: Euler-type non-homogeneous differential equations with three Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12(2), 205–234 (2009)MathSciNetMATH Kilbas, A.A., Zhukovskaya, N.V.: Euler-type non-homogeneous differential equations with three Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12(2), 205–234 (2009)MathSciNetMATH
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Zurück zum Zitat Kilbas, A.A., Zhukovskaya, N.V.: Solution in closed form linear inhomogeneous Euler type equations with fractional derivatives. Dokl. NAS Belarus 53(4), 30–36 (2009, in Russian) Kilbas, A.A., Zhukovskaya, N.V.: Solution in closed form linear inhomogeneous Euler type equations with fractional derivatives. Dokl. NAS Belarus 53(4), 30–36 (2009, in Russian)
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Zurück zum Zitat Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, vol. 301. Longman, Harlow (1994) Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, vol. 301. Longman, Harlow (1994)
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Zurück zum Zitat Kiryakova, V.: Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal. 2(4), 445–462 (1999)MathSciNetMATH Kiryakova, V.: Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal. 2(4), 445–462 (1999)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259 (2000)MathSciNetMATH Kiryakova, V.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259 (2000)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V.: Some special functions related to fractional calculus and fractional (non-integer) order control systems and equations. Facta Universitatis (Sci. J. Univ. Nis) Ser.: Autom. Control Robot. 7(1), 79–98 (2008) Kiryakova, V.: Some special functions related to fractional calculus and fractional (non-integer) order control systems and equations. Facta Universitatis (Sci. J. Univ. Nis) Ser.: Autom. Control Robot. 7(1), 79–98 (2008)
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Zurück zum Zitat Kiryakova, V.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59(5), 1128–1141 (2010)MathSciNetMATH Kiryakova, V.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59(5), 1128–1141 (2010)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V.: The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus. Comput. Math. Appl. 59(5), 1885–1895 (2010)MathSciNetMATH Kiryakova, V.: The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus. Comput. Math. Appl. 59(5), 1885–1895 (2010)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V., Al-Saqabi, B.N.: Solutions of Erdelyi–Kober fractional integral, differential and differintegral equations of second type. C. R. Acad. Bulg. Sci. 50(1), 27–30 (1997)MathSciNetMATH Kiryakova, V., Al-Saqabi, B.N.: Solutions of Erdelyi–Kober fractional integral, differential and differintegral equations of second type. C. R. Acad. Bulg. Sci. 50(1), 27–30 (1997)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V., Al-Saqabi, B.N.: Transmutation method for solving Erdelyi–Kober fractional differintegral equations. J. Math. Anal. Appl. 211(1), 347–364 (1997)MathSciNetMATH Kiryakova, V., Al-Saqabi, B.N.: Transmutation method for solving Erdelyi–Kober fractional differintegral equations. J. Math. Anal. Appl. 211(1), 347–364 (1997)MathSciNetMATH
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Zurück zum Zitat Kiryakova, V., Luchko, Yu.: The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis. In: AIP Conference Proceedings, AIP Publishing, Melville, NY, vol. 1301, pp. 597–613 (2010)MathSciNet Kiryakova, V., Luchko, Yu.: The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis. In: AIP Conference Proceedings, AIP Publishing, Melville, NY, vol. 1301, pp. 597–613 (2010)MathSciNet
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Zurück zum Zitat Kleinhans, D., Friedrich, R.: Continuous-time random walks: simulations of continuous trajectories. Phys. Rev. E 76, 061102/1–6 (2007) Kleinhans, D., Friedrich, R.: Continuous-time random walks: simulations of continuous trajectories. Phys. Rev. E 76, 061102/1–6 (2007)
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Zurück zum Zitat Kochubei, A.N.: Fractional order diffusion. J. Differ. Equ. 26, 485–492 (1990). English transl. from Russian Differ. Uravn. Kochubei, A.N.: Fractional order diffusion. J. Differ. Equ. 26, 485–492 (1990). English transl. from Russian Differ. Uravn.
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Zurück zum Zitat Kochubei, A.N.: General fractional calculus, evolution equations, and renewal processes. Integral Equ. Oper. Theory 71, 583–600 (2011)MathSciNetMATH Kochubei, A.N.: General fractional calculus, evolution equations, and renewal processes. Integral Equ. Oper. Theory 71, 583–600 (2011)MathSciNetMATH
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Zurück zum Zitat Krantz, S.: Complex Analysis: The Geometric Viewpoint, 2nd edn. Mathematical Association of America Inc., Washington, DC (2004) Krantz, S.: Complex Analysis: The Geometric Viewpoint, 2nd edn. Mathematical Association of America Inc., Washington, DC (2004)
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Zurück zum Zitat Kummer, E.E.: Über die hypergeometrische Reihe \(1 + \tfrac{\alpha \cdot \beta } {1\cdot \gamma }\ x + \tfrac{\alpha (\alpha +1)\beta (\beta +1)} {1\cdot 2\cdot \gamma (\gamma +1)} x^{2} + \tfrac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)} {1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)} x^{3} + \mbox{ etc.}\) J. für die reine und angewandte Mathematik 15, 39–83, 127–172 (1836, in German) Kummer, E.E.: Über die hypergeometrische Reihe \(1 + \tfrac{\alpha \cdot \beta } {1\cdot \gamma }\ x + \tfrac{\alpha (\alpha +1)\beta (\beta +1)} {1\cdot 2\cdot \gamma (\gamma +1)} x^{2} + \tfrac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)} {1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)} x^{3} + \mbox{ etc.}\) J. für die reine und angewandte Mathematik 15, 39–83, 127–172 (1836, in German)
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Zurück zum Zitat Kummer, E.E.: De integralibus quibusdam definitis et seriebus infinitis. J. für die reine und angewandte Mathematik 17, 228–242 (1837, in Latin) Kummer, E.E.: De integralibus quibusdam definitis et seriebus infinitis. J. für die reine und angewandte Mathematik 17, 228–242 (1837, in Latin)
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Zurück zum Zitat Laine I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Basel (1993) Laine I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Basel (1993)
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Zurück zum Zitat Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380-387 (1958)MathSciNetMATH Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380-387 (1958)MathSciNetMATH
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Zurück zum Zitat Laskin, N.: Fractional Poisson processes. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003)MathSciNetMATH Laskin, N.: Fractional Poisson processes. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003)MathSciNetMATH
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Zurück zum Zitat Laskin, N.: Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50, 113513/1–12 (2009)MathSciNet Laskin, N.: Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50, 113513/1–12 (2009)MathSciNet
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Zurück zum Zitat Le Roy, É.: Sur les séries divergentes et les fonctions définies par un développement de Taylor. Toulouse Ann. 2(2), 317–430 (1900)MATH Le Roy, É.: Sur les séries divergentes et les fonctions définies par un développement de Taylor. Toulouse Ann. 2(2), 317–430 (1900)MATH
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Zurück zum Zitat Letnikov, A.V.: Theory of differentiation with an arbitrary index. Math. Sb. 3, 1–66 (1868, in Russian) Letnikov, A.V.: Theory of differentiation with an arbitrary index. Math. Sb. 3, 1–66 (1868, in Russian)
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Zurück zum Zitat Letnikov, A.V.: On historical development of differentiation theory with an arbitrary index. Mat. Sb. 3, 85–112 (1868, in Russian) Letnikov, A.V.: On historical development of differentiation theory with an arbitrary index. Mat. Sb. 3, 85–112 (1868, in Russian)
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Zurück zum Zitat Letnikov, A.V., Chernykh V.A.: The Foundation of Fractional Calculus, Moscow, Neftegaz (2011) (in Russian) Letnikov, A.V., Chernykh V.A.: The Foundation of Fractional Calculus, Moscow, Neftegaz (2011) (in Russian)
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Zurück zum Zitat Levin, B.Ya.: Distribution of Zeros of Entire Functions. AMS, Providence, 2nd printing (1980). [First published in Russian by Nauka, Moscow (1956)] Levin, B.Ya.: Distribution of Zeros of Entire Functions. AMS, Providence, 2nd printing (1980). [First published in Russian by Nauka, Moscow (1956)]
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Zurück zum Zitat Lindelöf, E.: Sur la détermination de la croissance des fonctions entières définies par un developpment de Taylor. Bull. des Sciences Mathématigues (Ser. II) 27, 213–226 (1903) Lindelöf, E.: Sur la détermination de la croissance des fonctions entières définies par un developpment de Taylor. Bull. des Sciences Mathématigues (Ser. II) 27, 213–226 (1903)
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Zurück zum Zitat Liouville, J.: Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. J. l’École Roy. Polytéchn. 13(Sect. 21), 1–69 (1832) Liouville, J.: Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. J. l’École Roy. Polytéchn. 13(Sect. 21), 1–69 (1832)
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Zurück zum Zitat Liouville, J.: Mémoire sur le calcul des différentielles à indices quelconques. J. l’École Roy. Polytéchn. 13(Sect. 21), 71–162 (1832) Liouville, J.: Mémoire sur le calcul des différentielles à indices quelconques. J. l’École Roy. Polytéchn. 13(Sect. 21), 71–162 (1832)
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Zurück zum Zitat Liouville, J.: Mémoire sur l’intégration des équations différentielles à indices fractionnaires. J. l’École Roy. Polytéchn. 15(55), 55–84 (1837) Liouville, J.: Mémoire sur l’intégration des équations différentielles à indices fractionnaires. J. l’École Roy. Polytéchn. 15(55), 55–84 (1837)
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Zurück zum Zitat Luchko, Yu.: Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2, 463–488 (1999)MathSciNetMATH Luchko, Yu.: Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2, 463–488 (1999)MathSciNetMATH
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Zurück zum Zitat Luchko, Yu.: Asymptotics of zeros of the Wright function. J. Anal. Appl. (ZAA) 19, 583–596 (2000) Luchko, Yu.: Asymptotics of zeros of the Wright function. J. Anal. Appl. (ZAA) 19, 583–596 (2000)
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Zurück zum Zitat Luchko, Yu.: Algorithms for evaluation of the Wright function for the real arguments’ values. Fract. Calc. Appl. Anal. 11, 57–75 (2008)MathSciNetMATH Luchko, Yu.: Algorithms for evaluation of the Wright function for the real arguments’ values. Fract. Calc. Appl. Anal. 11, 57–75 (2008)MathSciNetMATH
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Zurück zum Zitat Luchko, Yu., Gorenflo, R.: Scale-invariant solutions of a partial differential equation of fractional order. Fract. Calc. Appl. Anal. 1, 63–78 (1998)MathSciNetMATH Luchko, Yu., Gorenflo, R.: Scale-invariant solutions of a partial differential equation of fractional order. Fract. Calc. Appl. Anal. 1, 63–78 (1998)MathSciNetMATH
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Zurück zum Zitat Luchko, Yu., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24, 207–233 (1999)MathSciNetMATH Luchko, Yu., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24, 207–233 (1999)MathSciNetMATH
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Zurück zum Zitat Luchko, Yu., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29, 73–85 (1995)MathSciNetMATH Luchko, Yu., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29, 73–85 (1995)MathSciNetMATH
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Zurück zum Zitat Mac Robert, T.M.: Proofs of some formulae for generalized hypergeometric function and certain related functions. Philos. Mag. 26, 82–93 (1938)MATH Mac Robert, T.M.: Proofs of some formulae for generalized hypergeometric function and certain related functions. Philos. Mag. 26, 82–93 (1938)MATH
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Zurück zum Zitat Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Connecticut (2006) Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Connecticut (2006)
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Zurück zum Zitat Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)MathSciNetMATH Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)MathSciNetMATH
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Zurück zum Zitat Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996)MathSciNetMATH Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996)MathSciNetMATH
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Zurück zum Zitat Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien/New York (1997) Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien/New York (1997)
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Zurück zum Zitat Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)MATH Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)MATH
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Zurück zum Zitat Mainardi, F., Gorenflo, R.: Fractional calculus: special functions and applications. In: Cocolicchio, D., et al. (eds.) Advanced Special Functions and Applications. Proceedings of the Workshop, Melfi, Italy, May 9–12, 1999. Rome: Aracne Editrice. Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics, vol. 1, 165–188 (2000)MathSciNet Mainardi, F., Gorenflo, R.: Fractional calculus: special functions and applications. In: Cocolicchio, D., et al. (eds.) Advanced Special Functions and Applications. Proceedings of the Workshop, Melfi, Italy, May 9–12, 1999. Rome: Aracne Editrice. Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics, vol. 1, 165–188 (2000)MathSciNet
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Zurück zum Zitat Mainardi, F., Gorenflo, R.: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 10, 269–308 (2007)MathSciNetMATH Mainardi, F., Gorenflo, R.: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 10, 269–308 (2007)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam. J. Math. 32(Spec. Iss.), 53–64 (2004) Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam. J. Math. 32(Spec. Iss.), 53–64 (2004)
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Zurück zum Zitat Mainardi, F., Gorenflo, R., Vivoli, A.: Renewal processes of Mittag-Leffler and Wright type. Fract. Calc. Appl. Anal. 8, 7–38 (2005)MathSciNetMATH Mainardi, F., Gorenflo, R., Vivoli, A.: Renewal processes of Mittag-Leffler and Wright type. Fract. Calc. Appl. Anal. 8, 7–38 (2005)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Gorenflo, R., Vivoli, A.: Beyond the Poisson renewal process: a tutorial survey. J. Comput. Appl. Math. 205, 725–735 (2007)MathSciNetMATH Mainardi, F., Gorenflo, R., Vivoli, A.: Beyond the Poisson renewal process: a tutorial survey. J. Comput. Appl. Math. 205, 725–735 (2007)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)MathSciNetMATH Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Pagnini, G.: Salvatore Pincherle: the pioneer of the Mellin–Barnes integrals. J. Comput. Appl. Math. 153, 331–342 (2003)MathSciNetMATH Mainardi, F., Pagnini, G.: Salvatore Pincherle: the pioneer of the Mellin–Barnes integrals. J. Comput. Appl. Math. 153, 331–342 (2003)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Pagnini, G., Saxena, R.K.: Fox H-functions in fractional diffusion. J. Comput. Appl. Math. 178, 321–331 (2005)MathSciNetMATH Mainardi, F., Pagnini, G., Saxena, R.K.: Fox H-functions in fractional diffusion. J. Comput. Appl. Math. 178, 321–331 (2005)MathSciNetMATH
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Zurück zum Zitat Mainardi, F., Raberto, M., Gorenflo, R., Scalas, E.: Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287, 468–481 (2000) Mainardi, F., Raberto, M., Gorenflo, R., Scalas, E.: Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287, 468–481 (2000)
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Zurück zum Zitat Mainardi, F., Rogosin, S.: The origin of infinitely divisible distributions: from de Finetti’s problem to Lévy-Khintchine formula. Math. Methods Econ. Finance 1, 37–55 (2006)MathSciNet Mainardi, F., Rogosin, S.: The origin of infinitely divisible distributions: from de Finetti’s problem to Lévy-Khintchine formula. Math. Methods Econ. Finance 1, 37–55 (2006)MathSciNet
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Zurück zum Zitat Malmquist, J.: Étude d’une fonction entière. Acta Math. 29, 203–215 (1905)MathSciNet Malmquist, J.: Étude d’une fonction entière. Acta Math. 29, 203–215 (1905)MathSciNet
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Zurück zum Zitat Mathai, A.M., Saxena, R.K.: The H-function with Applications in Statistics and Other Disciplines. Wiley Eastern Ltd, New Delhi (1978) Mathai, A.M., Saxena, R.K.: The H-function with Applications in Statistics and Other Disciplines. Wiley Eastern Ltd, New Delhi (1978)
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Zurück zum Zitat Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-function. Theory and applications. Springer, Dordrecht (2010) Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-function. Theory and applications. Springer, Dordrecht (2010)
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Zurück zum Zitat Meshkov, S.I.: Integral representation of fractional exponential functions and their application to dynamic problems in linear viscoelasticity. J. Appl. Mech. Tech. Phys. 11(1), 100–107 (1970); translation from Zh. Prikl. Mekh. i Techn. Fiz. 11(1), 103–110 (1970) Meshkov, S.I.: Integral representation of fractional exponential functions and their application to dynamic problems in linear viscoelasticity. J. Appl. Mech. Tech. Phys. 11(1), 100–107 (1970); translation from Zh. Prikl. Mekh. i Techn. Fiz. 11(1), 103–110 (1970)
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Zurück zum Zitat Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)MATH Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)MATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur l’intégrale de Laplace–Abel. C. R. Acad. Sci. Paris 135, 937–939 (1902)MATH Mittag-Leffler, M.G.: Sur l’intégrale de Laplace–Abel. C. R. Acad. Sci. Paris 135, 937–939 (1902)MATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Une généralization de l’intégrale de Laplace–Abel. Comp. Rend. Acad. Sci. Paris 136, 537–539 (1903)MATH Mittag-Leffler, M.G.: Une généralization de l’intégrale de Laplace–Abel. Comp. Rend. Acad. Sci. Paris 136, 537–539 (1903)MATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la nouvelle fonction E α (x). C. R. Acad. Sci. Paris 137, 554–558 (1903) Mittag-Leffler, M.G.: Sur la nouvelle fonction E α (x). C. R. Acad. Sci. Paris 137, 554–558 (1903)
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Zurück zum Zitat Mittag-Leffler, M.G.: Sopra la funzione E α (x). Rend. R. Acc. Lincei, (Ser. 5) 13, 3–5 (1904) Mittag-Leffler, M.G.: Sopra la funzione E α (x). Rend. R. Acc. Lincei, (Ser. 5) 13, 3–5 (1904)
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (première note). Acta Math. 23, 43–62 (1899)MathSciNetMATH Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (première note). Acta Math. 23, 43–62 (1899)MathSciNetMATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (seconde note). Acta Math. 24, 183–204 (1900)MathSciNetMATH Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (seconde note). Acta Math. 24, 183–204 (1900)MathSciNetMATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (troisème note). Acta Math. 24, 205–244 (1900)MathSciNet Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (troisème note). Acta Math. 24, 205–244 (1900)MathSciNet
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (quatrième note). Acta Math. 26, 353–392 (1902)MathSciNetMATH Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogène (quatrième note). Acta Math. 26, 353–392 (1902)MathSciNetMATH
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Zurück zum Zitat Mittag-Leffler, M.G.: Sur la representation analytique d’une branche uniforme d’une fonction monogène (cinquième note). Acta Math. 29, 101–181 (1905)MathSciNetMATH Mittag-Leffler, M.G.: Sur la representation analytique d’une branche uniforme d’une fonction monogène (cinquième note). Acta Math. 29, 101–181 (1905)MathSciNetMATH
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Zurück zum Zitat Nakhushev, A.M.: The Sturm–Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Dokl. Akad. Nauk SSSR. 234(2), 308–311 (1977, in Russian) Nakhushev, A.M.: The Sturm–Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Dokl. Akad. Nauk SSSR. 234(2), 308–311 (1977, in Russian)
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Zurück zum Zitat Nakhushev, A.M.: Fractional calculus and its applications. (Drobnoe ischislenie i ego primenenie.) Fizmatlit, Moskva (2003, in Russian) Nakhushev, A.M.: Fractional calculus and its applications. (Drobnoe ischislenie i ego primenenie.) Fizmatlit, Moskva (2003, in Russian)
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Zurück zum Zitat Olver (editor-in-chief), F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions, 951 + xv pp. and a CD. National Institute of Standards and Technology/Cambridge University Press, Gaithersburg/New York (2010) Olver (editor-in-chief), F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions, 951 + xv pp. and a CD. National Institute of Standards and Technology/Cambridge University Press, Gaithersburg/New York (2010)
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Zurück zum Zitat Nutting, P.G.: A general stress-strain-time formula. J. Frankl. Inst. 235, 513–524 (1943) Nutting, P.G.: A general stress-strain-time formula. J. Frankl. Inst. 235, 513–524 (1943)
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Zurück zum Zitat Pollard, H.: The completely monotonic character of the Mittag-Leffler function e α (−x). Bull. Am. Math. Soc. 54, 1115–1116 (1948)MathSciNetMATH Pollard, H.: The completely monotonic character of the Mittag-Leffler function e α (−x). Bull. Am. Math. Soc. 54, 1115–1116 (1948)MathSciNetMATH
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Zurück zum Zitat Polya, G.: Bemerkung über die Mittag-Lefflerschen Funktionen E 1∕α (x). Tôhoku Math. J. 19, 241–248 (1921)MATH Polya, G.: Bemerkung über die Mittag-Lefflerschen Funktionen E 1∕α (x). Tôhoku Math. J. 19, 241–248 (1921)MATH
[PolMan08]
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[Pop02]
Zurück zum Zitat Popov, A.Yu.: The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions. Diff. Equ. 38(5), 642–653 (2002); translation from Differ. Uravn. 38(5), 611–621 (2002) (English. Russian original) Popov, A.Yu.: The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions. Diff. Equ. 38(5), 642–653 (2002); translation from Differ. Uravn. 38(5), 611–621 (2002) (English. Russian original)
[Pop06]
Zurück zum Zitat Popov, A.Yu.: On a zeroes of Mittag-Leffler functions with parameter \(\rho < 1/2\). Anal. Math. 32(3), 207–246 (2006, in Russian) Popov, A.Yu.: On a zeroes of Mittag-Leffler functions with parameter \(\rho < 1/2\). Anal. Math. 32(3), 207–246 (2006, in Russian)
[PopSed03]
Zurück zum Zitat Popov, A.Yu., Sedletskii, A.M.: Zeros distribution of Mittag-Leffler functions. Dokl. Math. 67(3), 336–339 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk. 390(2), 165–168 (2003) Popov, A.Yu., Sedletskii, A.M.: Zeros distribution of Mittag-Leffler functions. Dokl. Math. 67(3), 336–339 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk. 390(2), 165–168 (2003)
[PopSed11]
Zurück zum Zitat Popov, A.Yu., Sedletskii, A.M.: Zeros distribution of Mittag-Leffler functions. Contemp. Math. Fundam. Directions. 40, 3–171 (2011,in Russian); translation in J. Math. Sci. 190, 209–409 (2013) Popov, A.Yu., Sedletskii, A.M.: Zeros distribution of Mittag-Leffler functions. Contemp. Math. Fundam. Directions. 40, 3–171 (2011,in Russian); translation in J. Math. Sci. 190, 209–409 (2013)
[Pos19]
Zurück zum Zitat Post, E.L.: Discussion of the solution of \((\mathrm{d}/\mathrm{d}x)^{1/2}y = y/x\) (problem ♯ 433). Am. Math. Mon. 26, 37–39 (1919)MathSciNet Post, E.L.: Discussion of the solution of \((\mathrm{d}/\mathrm{d}x)^{1/2}y = y/x\) (problem 433). Am. Math. Mon. 26, 37–39 (1919)MathSciNet
[Pra71]
Zurück zum Zitat Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)MathSciNetMATH Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)MathSciNetMATH
[PrBrMa-V2]
Zurück zum Zitat Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integral and Series. Special Functions, vol. 2. Gordon and Breach, New York (1986) Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integral and Series. Special Functions, vol. 2. Gordon and Breach, New York (1986)
[PrBrMa-V3]
Zurück zum Zitat Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integral and Series. More Special Functions, vol. 3. Gordon and Breach, New York (1990) Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integral and Series. More Special Functions, vol. 3. Gordon and Breach, New York (1990)
[Psk05]
Zurück zum Zitat Pskhu, A.V.: On real zeros of a Mittag-Leffler-type function. Math. Notes. 77(4), 592–599 (2005) [translation from Russian Matematicheskie Zametki] Pskhu, A.V.: On real zeros of a Mittag-Leffler-type function. Math. Notes. 77(4), 592–599 (2005) [translation from Russian Matematicheskie Zametki]
[Psk06]
Zurück zum Zitat Pskhu, A.V.: Partial Differential Equations of Fractional Order. Nauka, Moscow (2006, in Russian) Pskhu, A.V.: Partial Differential Equations of Fractional Order. Nauka, Moscow (2006, in Russian)
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Zurück zum Zitat Rabotnov, Yu.N.: On the Use of Singular Operators in the Theory of Viscoelasticity, Moscow, p. 50 (1973). Unpublished Lecture Notes for the CISM course on Rheology held in Udine, Oct 1973. http://www.cism.it Rabotnov, Yu.N.: On the Use of Singular Operators in the Theory of Viscoelasticity, Moscow, p. 50 (1973). Unpublished Lecture Notes for the CISM course on Rheology held in Udine, Oct 1973. http://​www.​cism.​it
[Rab74]
Zurück zum Zitat Rabotnov, Yu.N.: Experimental Evidence of the Principle of Hereditary in Mechanics of Solids, Moscow, p. 80 (1974). Unpublished Lecture Notes for the CISM course on Experimental Methods in Mechanics, A) Rheology, held in Udine, 24–29 Oct 1974. http://www.cism.it Rabotnov, Yu.N.: Experimental Evidence of the Principle of Hereditary in Mechanics of Solids, Moscow, p. 80 (1974). Unpublished Lecture Notes for the CISM course on Experimental Methods in Mechanics, A) Rheology, held in Udine, 24–29 Oct 1974. http://​www.​cism.​it
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Zurück zum Zitat Rabotnov, Yu.N., Papernik, L.H., Zvonov, E.N.: Tables of a Fractional-Exponential Function of Negative Parameter and Its Integral. Moscow, Nauka (1969, in Russian) Rabotnov, Yu.N., Papernik, L.H., Zvonov, E.N.: Tables of a Fractional-Exponential Function of Negative Parameter and Its Integral. Moscow, Nauka (1969, in Russian)
[Rai71]
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[RuDiKi96]
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Zurück zum Zitat Saichev, A., Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)MathSciNetMATH Saichev, A., Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)MathSciNetMATH
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Zurück zum Zitat Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25(2), 319–327 (1988)MathSciNetMATH Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25(2), 319–327 (1988)MathSciNetMATH
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Zurück zum Zitat Saxena, R.K.: Certain properties of generalized Mittag-Leffler function. In: Proceedings of the 3rd Annual Conference of the Society for Special Functions and Their Applications, Chennai, pp. 77–81 (2002) Saxena, R.K.: Certain properties of generalized Mittag-Leffler function. In: Proceedings of the 3rd Annual Conference of the Society for Special Functions and Their Applications, Chennai, pp. 77–81 (2002)
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Zurück zum Zitat Saxena, R.K.: In memorium of Charles Fox. Frat. Calc. Appl. Anal. 12(3), 337–344 (2009)MATH Saxena, R.K.: In memorium of Charles Fox. Frat. Calc. Appl. Anal. 12(3), 337–344 (2009)MATH
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Zurück zum Zitat Saxena R.K., Pogány T.K., Ram J. and Daiya J.: Dirichlet averages of generalized multi-index Mittag-Leffler functions. Am J. Math. 3(4), 174–187 (2010) Saxena R.K., Pogány T.K., Ram J. and Daiya J.: Dirichlet averages of generalized multi-index Mittag-Leffler functions. Am J. Math. 3(4), 174–187 (2010)
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Zurück zum Zitat Saxena, R.K., Saigo, M.: Certain properties of fractional calculus operators associated with generalized Wright function. Fract. Calc. Appl. Anal. 6, 141–154 (2005)MathSciNet Saxena, R.K., Saigo, M.: Certain properties of fractional calculus operators associated with generalized Wright function. Fract. Calc. Appl. Anal. 6, 141–154 (2005)MathSciNet
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Zurück zum Zitat Scalas, E., Gorenflo, R., Mainardi, F. and Raberto, M.: Revisiting the derivation of the fractional diffusion equation. Fractals 11(Suppl. S), 281–289 (2003) Scalas, E., Gorenflo, R., Mainardi, F. and Raberto, M.: Revisiting the derivation of the fractional diffusion equation. Fractals 11(Suppl. S), 281–289 (2003)
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Zurück zum Zitat Sedletskii, A.M.: Non-asymptotic properties of roots of a Mittag-Leffler type function. Math. Notes. 75(3), 372–386 (2004) [translation from Russian Matematicheskie Zametki] Sedletskii, A.M.: Non-asymptotic properties of roots of a Mittag-Leffler type function. Math. Notes. 75(3), 372–386 (2004) [translation from Russian Matematicheskie Zametki]
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Zurück zum Zitat Sedletskii, A.M.: Asymptotics of zeros of the Mittag-Leffler type function of order 1/2. Vestn. Mosk. Univ., Ser. I, (1), 22–28 (2007) (Russian, English); translation in Mosc. Univ. Math. Bull. 62(1), 22–28 (2007) Sedletskii, A.M.: Asymptotics of zeros of the Mittag-Leffler type function of order 1/2. Vestn. Mosk. Univ., Ser. I, (1), 22–28 (2007) (Russian, English); translation in Mosc. Univ. Math. Bull. 62(1), 22–28 (2007)
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Metadaten
Titel
Introduction
verfasst von
Rudolf Gorenflo
Anatoly A. Kilbas
Francesco Mainardi
Sergei V. Rogosin
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-43930-2_1

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