Fractional Calculus Questionnaire

Springer Monographs in Mathematics
Rudolf Gorenflo
Anatoly A. Kilbas
Francesco Mainardi
Sergei V. Rogosin
Mittag-Leffler
Functions,
Related Topics
and Applications
Springer Monographs in Mathematics
More information about this series at
http://www.springer.com/series/3733
Rudolf Gorenflo ? Anatoly A. Kilbas ?
Francesco Mainardi ? Sergei V. Rogosin
Mittag-Leffler Functions,
Related Topics
and Applications
123
Rudolf Gorenflo
Free University Berlin Mathematical
Institute
Berlin
Germany
Anatoly A. Kilbas (July 20, 1948 – June 28,
2010)
Belarusian State University Department
of Mathematics and Mechanics
Minsk
Belarus
Francesco Mainardi
University of Bologna Department
of Physics
Bologna
Italy
Sergei V. Rogosin
Belarusian State University Department
of Economics
Minsk
Belarus
ISSN 1439-7382
ISSN 2196-9922 (electronic)
Springer Monographs in Mathematics
ISBN 978-3-662-43929-6
ISBN 978-3-662-43930-2 (eBook)
DOI 10.1007/978-3-662-43930-2
Springer Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014949196
Mathematics Subject Classification: 33E12, 26A33, 34A08, 45K05, 44Axx, 60G22
? Springer-Verlag Berlin Heidelberg 2014
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To the memory of our colleague and friend
Anatoly Kilbas
Preface
The study of the Mittag-Leffler function and its various generalizations has become
a very popular topic in mathematics and its applications. However, during the twentieth century, this function was practically unknown to the majority of scientists,
since it was ignored in most common books on special functions. As a noteworthy
exception the handbook Higher Transcendental Functions, vol. 3, by A. Erdelyi
et al. deserves to be mentioned.
Now the Mittag-Leffler function is leaving its isolated role as Cinderella (using
the term coined by F.G. Tricomi for the incomplete gamma function).
The recent growing interest in this function is mainly due to its close relation
to the Fractional Calculus and especially to fractional problems which come from
applications.
Our decision to write this book was motivated by the need to fill the gap in the
literature concerning this function, to explain its role in modern pure and applied
mathematics, and to give the reader an idea of how one can use such a function in
the investigation of modern problems from different scientific disciplines.
This book is a fruit of collaboration between researchers in Berlin, Bologna and
Minsk. It has highly profited from visits of SR to the Department of Physics at
the University of Bologna and from several visits of RG to Bologna and FM to
the Department of Mathematics and Computer Science at Berlin Free University
under the European ERASMUS exchange. RG and SR appreciate the deep scientific
atmosphere at the University of Bologna and the perfect conditions they met there
for intensive research.
We are saddened that our esteemed and always enthusiastic co-author Anatoly
A. Kilbas is no longer with us, having lost his life in a tragic accident on 28 June
2010 in the South of Russia. We will keep him, and our inspiring joint work with
him, in living memory.
Berlin, Germany
Bologna, Italy
Minsk, Belarus
March 2014
Rudolf Gorenflo
Francesco Mainardi
Sergei Rogosin
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1
2 Historical Overview of the Mittag-Leffler Functions . . . . . . . . . . . . . . . . . . . .
2.1 A Few Biographical Notes on G?sta Magnus Mittag-Leffler . . . . . . .
2.2 The Contents of the Five Papers by Mittag-Leffler
on New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Further History of Mittag-Leffler Functions . . . . .. . . . . . . . . . . . . . . . . . . .
7
7
9
12
3 The Classical Mittag-Leffler Function .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Relations to Elementary and Special Functions … . . . . . . . . . . . . . . . . . . .
3.3 Recurrence and Differential Relations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4 Integral Representations and Asymptotics . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5 Distribution of Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6 Further Analytic Properties . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7 The Mittag-Leffler Function of a Real Variable … . . . . . . . . . . . . . . . . . . .
3.7.1 Integral Transforms . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.2 The Complete Monotonicity Property . .. . . . . . . . . . . . . . . . . . . .
3.7.3 Relation to Fractional Calculus. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.8 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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4 The Two-Parametric Mittag-Leffler Function . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Series Representation and Properties of Coefficients . . . . . . . . . . . . . . . .
4.2 Explicit Formulas: Relations to Elementary and Special
Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3 Differential and Recurrence Relations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4 Integral Relations and Asymptotics .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.5 The Two-Parametric Mittag-Leffler Function
as an Entire Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.6 Distribution of Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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4.7
4.8
4.9
4.10
Computations with the Two-Parametric Mittag-Leffler Function . . .
Extension for Negative Values of the First Parameter .. . . . . . . . . . . . . .
Further Analytic Properties . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The Two-Parametric Mittag-Leffler Function of a Real Variable . . .
4.10.1 Integral Transforms of the Two-Parametric
Mittag-Leffler Function .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.10.2 The Complete Monotonicity Property . .. . . . . . . . . . . . . . . . . . . .
4.10.3 Relations to the Fractional Calculus . . . .. . . . . . . . . . . . . . . . . . . .
4.11 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
74
80
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5 Mittag-Leffler Functions with Three Parameters . . . .. . . . . . . . . . . . . . . . . . . .
5.1 The Prabhakar (Three-Parametric Mittag-Leffler) Function .. . . . . . . .
5.1.1 Definition and Basic Properties . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1.2 Integral Representations and Asymptotics .. . . . . . . . . . . . . . . . .
5.1.3 Integral Transforms of the Prabhakar Function.. . . . . . . . . . . .
5.1.4 Fractional Integrals and Derivatives of the
Prabhakar Function . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1.5 Relations to the Wright Function, H -Function
and Other Special Functions.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Generalized (Kilbas?Saigo) Mittag-Leffler Type Functions .. . . . . . . .
5.2.1 Definition and Basic Properties . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.2 The Order and Type of the Entire Function E?;m;l .z/ . . . . . .
5.2.3 Recurrence Relations for E?;m;l .z/ . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.4 Connection of En;m;l .z/ with Functions
of Hypergeometric Type .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.5 Differentiation Properties of En;m;l .z/ . .. . . . . . . . . . . . . . . . . . . .
5.2.6 Fractional Integration of the Generalized
Mittag-Leffler Function .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.7 Fractional Differentiation of the Generalized
Mittag-Leffler Function .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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101
6 Multi-index Mittag-Leffler Functions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 The Four-Parametric Mittag-Leffler Function:
Luchko?Kilbas?Kiryakova?s Approach . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.1 Definition and Special Cases . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.3 Integral Representations and Asymptotics .. . . . . . . . . . . . . . . . .
6.1.4 Extended Four-Parametric Mittag-Leffler Functions .. . . . . .
6.1.5 Relations to the Wright Function and the H-Function . . . . .
6.1.6 Integral Transforms of the Four-Parametric
Mittag-Leffler Function .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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6.1.7
6.2
6.3
6.4
Integral Transforms with the Four-Parametric
Mittag-Leffler Function in the Kernel . . .. . . . . . . . . . . . . . . . . . . .
6.1.8 Relations to the Fractional Calculus . . . .. . . . . . . . . . . . . . . . . . . .
Mittag-Leffler Functions with 2n Parameters . . . .. . . . . . . . . . . . . . . . . . . .
6.2.1 Definition and Basic Properties . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.2 Representations in Terms of Hypergeometric Functions .. .
6.2.3 Integral Representations and Asymptotics .. . . . . . . . . . . . . . . . .
6.2.4 Extension of the 2n-Parametric Mittag-Leffler
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.5 Relations to the Wright Function and to the
H-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.6 Integral Transforms with the Multi-parametric
Mittag-Leffler Functions .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.7 Relations to the Fractional Calculus . . . .. . . . . . . . . . . . . . . . . . . .
Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7 Applications to Fractional Order Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Fractional Order Integral Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1.1 The Abel Integral Equation .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1.2 Other Integral Equations Whose Solutions
Are Represented via Generalized Mittag-Leffler
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2 Fractional Ordinary Differential Equations.. . . . . .. . . . . . . . . . . . . . . . . . . .
7.2.1 Fractional Ordinary Differential Equations
with Constant Coefficients.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2.2 Ordinary FDEs with Variable Coefficients . . . . . . . . . . . . . . . . .
7.2.3 Other Types of Ordinary Fractional Differential
Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3 Differential Equations with Fractional Partial Derivatives . . . . . . . . . .
7.3.1 Cauchy-Type Problems for Differential
Equations with Fractional Partial Derivatives . . . . . . . . . . . . . .
7.3.2 The Cauchy Problem for Differential Equations
with Fractional Partial Derivatives . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8 Applications to Deterministic Models . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1 Fractional Relaxation and Oscillations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1.1 Simple Fractional Relaxation and Oscillation . . . . . . . . . . . . . .
8.1.2 The Composite Fractional Relaxation and Oscillations . . . .
8.2 Examples of Applications of the Fractional Calculus
in Physical Models.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.1 Linear Visco-Elasticity . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.2 Other Deterministic Fractional Models .. . . . . . . . . . . . . . . . . . . .
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8.3
Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.1 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.2 Notes on Fractional Differential Equations .. . . . . . . . . . . . . . . .
8.3.3 Notes on the Fractional Calculus in Linear
Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
227
227
228
9 Applications to Stochastic Models . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.2 The Mittag-Leffler Process According to Pillai . .. . . . . . . . . . . . . . . . . . . .
9.3 Elements of Renewal Theory and Continuous Time
Random Walk (CTRW) . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.3.1 Renewal Processes . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.3.2 Continuous Time Random Walk (CTRW) . . . . . . . . . . . . . . . . . .
9.3.3 The Renewal Process as a Special CTRW . . . . . . . . . . . . . . . . . .
9.4 The Poisson Process and Its Fractional Generalization
(the Renewal Process of Mittag-Leffler Type) . . .. . . . . . . . . . . . . . . . . . . .
9.4.1 The Mittag-Leffler Waiting Time Density . . . . . . . . . . . . . . . . . .
9.4.2 The Poisson Process . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.4.3 The Renewal Process of Mittag-Leffler Type .. . . . . . . . . . . . . .
9.4.4 Thinning of a Renewal Process. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5 The Fractional Diffusion Process . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5.1 Renewal Process with Reward . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5.2 Limit of the Mittag-Leffler Renewal Process .. . . . . . . . . . . . . .
9.5.3 Subordination in the Space-Time Fractional
Diffusion Equation .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5.4 The Rescaling and Respeeding Concept
Revisited: Universality of the Mittag-Leffler Density .. . . . .
9.6 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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A The Eulerian Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.1 Analytic Continuation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.2 The Graph of the Gamma Function on the Real Axis . . . . . .
A.1.3 The Reflection or Complementary Formula . . . . . . . . . . . . . . . .
A.1.4 The Multiplication Formulas . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.5 Pochhammer?s Symbols . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.6 Hankel Integral Representations . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.7 Notable Integrals via the Gamma Function .. . . . . . . . . . . . . . . .
A.1.8 Asymptotic Formulas . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1.9 Infinite Products .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2.1 Euler?s Integral Representation .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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8.4
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A.2.3 Trigonometric Integral Representation… . . . . . . . . . . . . . . . . . . .
A.2.4 Relation to the Gamma Function .. . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2.5 Other Integral Representations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2.6 Notable Integrals via the Beta Function . . . . . . . . . . . . . . . . . . . .
Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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B The Basics of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.1 Definition and Series Representations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.2 Growth of Entire Functions: Order, Type and Indicator Function .. .
B.3 Weierstrass Canonical Representation: Distribution of Zeros . . . . . . .
B.4 Entire Functions of Completely Regular Growth . . . . . . . . . . . . . . . . . . . .
B.5 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
285
285
286
287
289
290
292
C Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.1 Fourier Type Transforms . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.2 The Laplace Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.3 The Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.4 Simple Examples and Tables of Transforms of Basic
Elementary and Special Functions .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.5 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
299
299
304
307
311
313
314
D The Mellin?Barnes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
D.1 Definition: Contour of Integration . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
D.2 Asymptotic Methods for the Mellin?Barnes Integral . . . . . . . . . . . . . . . .
D.3 Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
D.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
319
319
322
324
325
E Elements of Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.1 Introduction to the Riemann?Liouville Fractional Calculus. . . . . . . . .
E.2 The Liouville?Weyl Fractional Calculus . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.3 The Abel?Riemann Fractional Calculus . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.3.1 The Abel?Riemann Fractional Integrals and
Derivatives.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.4 The Caputo Fractional Calculus . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.4.1 The Caputo Fractional Derivative . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.5 The Riesz?Feller Fractional Calculus . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
E.5.1 The Riesz Fractional Integrals and Derivatives .. . . . . . . . . . . .
E.5.2 The Feller Fractional Integrals and Derivatives . . . . . . . . . . . .
E.6 The Gr?nwald?Letnikov Fractional Calculus . . . .. . . . . . . . . . . . . . . . . . . .
E.6.1 The Gr?nwald?Letnikov Approximation
in the Riemann?Liouville Fractional Calculus . . . . . . . . . . . . .
327
327
330
334
A.3
A.4
334
336
336
338
339
341
344
345
xiv
Contents
E.6.2
E.7
The Gr?nwald?Letnikov Approximation
in the Riesz?Feller Fractional Calculus .. . . . . . . . . . . . . . . . . . . . 348
Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 348
F Higher Transcendental Functions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.1
Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.1.1
Classical Gauss Hypergeometric Functions . . . . . . . . . . . . . . . .
F.1.2
Euler Integral Representation: Mellin?Barnes
Integral Representation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.1.3
Basic Properties of Hypergeometric Functions .. . . . . . . . . . . .
F.1.4
The Hypergeometric Differential Equation .. . . . . . . . . . . . . . . .
F.1.5
Kummer?s and Tricomi?s Confluent
Hypergeometric Functions . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.1.6
Generalized Hypergeometric Functions
and Their Properties . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.2
Wright Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.2.1
The Classical Wright Function . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.2.2
Mellin?Barnes Integral Representation and
Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.2.3
The Bessel?Wright Function: Generalized
Wright Functions and Fox?Wright Functions . . . . . . . . . . . . . .
F.3
Meijer G-Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.3.1
Definition via Integrals: Existence . . . . . .. . . . . . . . . . . . . . . . . . . .
F.3.2
Basic Properties of the Meijer G-Functions . . . . . . . . . . . . . . . .
F.3.3
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.3.4
Relations to Fractional Calculus . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.3.5
Integral Transforms of G-Functions . . . .. . . . . . . . . . . . . . . . . . . .
F.4
Fox H -Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.4.1
Definition via Integrals: Existence . . . . . .. . . . . . . . . . . . . . . . . . . .
F.4.2
Series Representations and Asymptotics:
Recurrence Relations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.4.3
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.4.4
Relations to Fractional Calculus . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.4.5
Integral Transforms of H -Functions .. . .. . . . . . . . . . . . . . . . . . . .
F.5
Historical and Bibliographical Notes . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
F.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
351
351
351
353
354
355
356
360
361
361
362
363
365
365
366
367
368
369
370
370
374
377
379
381
382
385
References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 389
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 441
Chapter 1
Introduction
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.
The function E? .z/ is named after the great Swedish mathematician G?sta
Magnus Mittag-Leffler (1846?1927) who defined it by a power series
E? .z/ D
1
X
kD0
zk
;
.?k C 1/
? 2 C; Re ? > 0;
(1.0.1)
and studied its properties in 1902?1905 in five subsequent notes [ML1, ML2, ML3,
ML4, ML5-5] in connection with his summation method for divergent series.
This function provides a simple generalization of the exponential function
because of the replacement of k? D .k C 1/ by .?k/? D .?k C 1/ in the
denominator of the power terms of the exponential series.
During the first half of the twentieth century the Mittag-Leffler function remained
almost unknown to the majority of scientists. They unjustly ignored it in many
treatises on special functions, including the most common (Abramowitz and Stegun
[AbrSte72] and its novel version ?NIST Handbook of Mathematical Functions?
[NIST]). Furthermore, there appeared some relevant works where the authors
arrived at series or integral representations of this function without recognizing
it, e.g. (Gnedenko and Kovalenko [GneKov68]), (Balakrishnan [BalV85]) and
(Sanz-Serna [San88]). A description of the most important properties of this
function is present in the third volume [Bat-3] of the Handbook on Higher
Transcendental Functions of the Bateman Project, (Erdelyi et al.). In it, the authors
have included the Mittag-Leffler functions in their Chapter XVIII devoted to the
so-called miscellaneous functions. The attribution of ?miscellaneous? to the MittagLeffler function is due to the fact that it was only later, in the 1960s, that it was
recognized to belong to a more general class of higher transcendental functions,
known as Fox H -functions (see, e.g., [MatSax78, KilSai04, MaSaHa10]). In fact,
? Springer-Verlag Berlin Heidelberg 2014
R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications,
Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__1
1
2
1 Introduction
this class was well-established only after the seminal paper by Fox [Fox61]. A more
detailed account of the Mittag-Leffler function is given in the treatise on complex
functions by Sansone and Gerretsen [SanGer60]. However, the most specialized
treatise, where more details on the functions of Mittag-Leffler type are given, is
surely the book by Dzherbashyan [Dzh66], in Russian. Unfortunately, no official
English translation of this book is presently available. Nevertheless, Dzherbashyan
has done a lot to popularize the Mittag-Leffler function from the point of view of its
special role among entire functions of a complex variable, where this function can
be considered as the simplest non-trivial generalization of the exponential function.
Successful applications of the Mittag-Leffler function and its generalizations, and
their direct involvement in problems of physics, biology, chemistry, engineering
and other applied sciences in recent decades has made them better known among
scientists. A considerable literature is devoted to the investigation of the analyticity
properties of these functions; in the references we quote several authors who, after
Mittag-Leffler, have investigated such functions from a mathematical point of view.
At last, the 2000 Mathematics Subject Classification has included these functions in
item 33E12: ?Mittag-Leffler functions and generalizations?.
Starting from the classical paper of Hille and Tamarkin [HilTam30] in which the
solution of Abel integral equation of the second kind

.x/
.?/
Zx
.t/
dt D f .x/; 0 < ? < 1; 0 < x < 1;
.x t/1?
(1.0.2)
0
is presented in terms of the Mittag-Leffler function, this function has become very
important in the study of different types of integral equations. We should also
mention the 1954 paper by Barret [Barr54] which was concerned with the general
solution of the linear fractional differential equation with constant coefficients.
But the real importance of this function was recognized when its special role
in Fractional Calculus was discovered (see, e.g., [SaKiMa93]). In recent times the
attention of mathematicians and applied scientists towards the functions of MittagLeffler type has increased, overall because of their relation to the fractional calculus
and its applications. Because the fractional calculus has attracte