The Wright functions of the second kind in Mathematical Physics
TTHE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS
FRANCESCO MAINARDI AND ARMANDO CONSIGLIO abstract In this review paper we stress the importance of the higher transcenden-tal Wright functions of the second kind in the framework of Mathemati-cal Physics. We first start with the analytical properties of the classicalWright functions of which we distinguish two kinds. We then justifythe relevance of the Wright functions of the second kind as fundamen-tal solutions of the time-fractional diffusion-wave equations. Indeed,we think that this approach is the most accessible point of view.fordescribing Non-Gaussian stochastic processes and the transition fromsub-diffusion processes to wave propagation. Through the sections of thetext and suitable appendices we plan to address the reader in this path-way towards the applications of the Wright functions of the second kind.
Keywords : Fractional Calculus, Wright Functions, Green’s Functions,Diffusion-Wave Equation, Laplace Transform.
MSC : 26A33, 33E12, 34A08, 34C26.2.
Introduction
The special functions play a fundamental role in all fields of AppliedMathematics and Mathematical Physics because any analytical resultsare expressed in terms of some of these functions. Even if the topic ofspecial functions can appear boring and their properties mainly treatedin handbooks, we would promote the relevance of some of them notyet so well known. We devote our attention to the Wright functions,in particular with the class of the second kind. These functions, aswe will see hereafter, are fundamental to deal with some non-standarddeterministic and stochastic processes. Indeed the Gaussian function(known as the normal probability distribution) must be generalizedin a suitable way in the framework of partial differential equations ofnon-integer order.This work is organized as follows. In Section 2 we introduce the Wrightfunctions, entire in the complex plane that we distinguish in two kindsin relation on the value-range of the two parameters on which they
PAPER PUBLISHED IN MATHEMATICS (MDPI), VOL 8 NO 6 (2020),884/26PP. DOI: 10.3390/MATH8060884 a r X i v : . [ m a t h . G M ] J u l FRANCESCO MAINARDI AND ARMANDO CONSIGLIO depend. In particular we devote our attention on two Wright functionsof the second kind introduced by Mainardi with the term of auxiliaryfunctions. One of them, known as M-Wright function generalizes theGaussian function so it is expected to play a fundamental role in non-Gaussian stochastic processes.Indeed In Section 3 we show how the Wright functions of the second kindare relevant in the analysis of time-fractional diffusion and diffusion-waveequations being related to their fundamental solutions. This analysisleads to generalize the known results r of the standard diffusion equationin the one-dimensional case, that is recalled in Appendix A by meansof auxiliary functions as particular cases of the Wright functions of thesecond kind known as M-Wright or Mainardi functions. For readers’convenience, in Appendix B we will also provide a introduction to thetime-derivative of fractional order in the Caputo sense We remind thatnowadays, as usual, by fractional order we mean a non-integer order,sothat the term ” fractional ” is a misnomer kept only for historical reasons.In Section 4 we consider again the Mainsrdi auxiliary functions functionsfor their role in probability theory and in particular in the framework ofL´evy stable distributions whose general theory is recalled in AppendixC.In Section 5 we show how the auxiliary functions turn out to be includedin a class that we denote the four sister functions . On their turn thesefour functions depending on a real parameter ν ∈ (0 ,
1) are the naturalgeneralization of the three sisters functions introduced in Appendix Adevoted to the standard diffusion equation. The attribute of sisters wasput by one of us (F. M.) because of their inter-relations, in his lecturenotes on Mathematical Physics, so it has only a personal reason thatwe hope to be shared by the readers.Finally, in Section 6, we provide some concluding remarks payingattention to work to be done in the next future.We point out that we have equipped our theoretical analysis with severalplots hoping they will be considered illuminating for the interestedreaders. We also note that we have limited our review to the simplestboundary values problems of equations in one space dimension referringthe readers to suitable references for more general treatments in Section3.1.3.
The Wright functions of the second kind and theMainardi auxiliary functions
The classical
Wright function , that we denote by W λ,µ ( z ), is defined bythe series representation convergent in the whole complex plane,(1) W λ,µ ( z ) := ∞ (cid:88) n =0 z n n !Γ( λn + µ ) , λ > − , µ ∈ C , HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS The integral representation reads as:(2) W λ,µ ( z ) = 12 πi (cid:90) Ha − e σ + zσ − λ dσσ µ , λ > − , µ ∈ C , where Ha − denotes the Hankel path: this one is a loop which startsfrom −∞ along the lower side of negative real axis, encircles with asmall circle the axes origin and ends at −∞ along the upper side of thenegative real axis. W λ,µ ( z ) is then an entire function for all λ ∈ ( − , + ∞ ). OriginallyWright assumed λ ≥ − < λ <
0, [50]. We note that in the Vol 3, Chapter 18 of thehandbook of the Bateman Project [10], presumably for a misprint,the parameter λ is restricted to be non-negative, whereas the Wrightfunctions remained practically ignored in other handbooks. In 1993Mainardi, being aware only of the Bateman handbook, proved that theWright function is entire also for − < λ < z = r > r can bethe time variable t or the space variable x ) the Wright functions aredistinguished in first kind ( λ ≥
0) and second kind ( − < λ <
0) asoutlined in the Appendix F of the book by Mainardi [35]. In particular,for the asymptotic behaviour, we refer the interested reader to the twopapers by Wong and Zhao [46, 47], and to the surveys by Luchko andby Paris in the Handbook of Fractional Calculus and Applications, seerespectively [25], [41], and references therein.We note that the Wright functions are entire of order 1 / (1 + λ ) henceonly the first kind functions ( λ ≥
0) are of exponential order whereasthe second kind functions ( − < λ <
0) are not of exponential order.The case λ = 0 is trivial since W ,µ ( z ) = e z / Γ( µ ) . As a consequence ofthe distinction in the kinds, we must point out the different Laplacetransforms proved e.g. in [15],[35], see also the recent survey on Wrightfunctions by Luchko [25]. We have:for the first kind, when λ ≥ W λ,µ ( ± r ) ÷ s E λ,µ (cid:18) ± s (cid:19) ;for the second kind, when − < λ < ν = − λ so 0 < ν < W − ν,µ ( − r ) ÷ E ν,µ + ν ( − s ) . FRANCESCO MAINARDI AND ARMANDO CONSIGLIO Above we have introduced the Mittag-Leffler function in two parameters α > β ∈ C defined as its convergent series for all z ∈ C(5) E α,β ( z ) := ∞ (cid:88) n =0 z n Γ( αn + β ) . For more details on the special functions of the Mittag-leffler type werefer the interested readers to the treatise by Gorenflo et al [14], wherein the forthcoming 2-nd edition also the Wright functions are treatedin some detail.In particular, two Wright functions of the second kind, originally intro-duced by Mainardi and named F ν ( z ) and M ν ( z ) (0 < ν < auxiliary functions in virtue of their role in the time fractional diffusionequations considered in the next section. These functions, F ν ( z ) and M ν ( z ), are indeed special cases of the Wright function of the secondkind W λ,µ ( z ) by setting, respectively, λ = − ν and µ = 0 or µ = 1 − ν .Hence we have:(6) F ν ( z ) := W − ν, ( − z ) , < ν < , and(7) M ν ( z ) := W − ν, − ν ( − z ) , < ν < , Those functions are interrelated through the following relation:(8) F ν ( z ) = νzM ν ( z ) , which reminds us the second relation in 7, seen for the standard diffusionequation.The series representations of the auxiliary functions are derived fromthose of W λ,µ ( z ). Then:(9) F ν ( z ) := ∞ (cid:88) n =1 ( − z ) n n !Γ( − νn ) = 1 π ∞ (cid:88) n =1 ( − z ) n − n ! Γ( νn + 1) sin ( πνn ) , and(10) M ν ( z ) := ∞ (cid:88) n =0 ( − z ) n n !Γ[ − νn + (1 − ν )] = 1 π ∞ (cid:88) n =1 ( − z ) n − ( n − νn ) sin ( πνn ) , where it has been used in both cases the reflection formula for theGamma function (Eq. 11) among the first and the second step of Eqs.(9) and (10),(11) Γ( ζ )Γ(1 − ζ ) = π/ sin πζ. Also the integral representations of the auxiliary functions are derivedfrom those of W λ,µ ( z ). Then:(12) F ν ( z ) := 12 πi (cid:90) Ha − e σ − zσ ν dσ, z ∈ C , < ν < HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS and(13) M ν ( z ) := 12 πi (cid:90) Ha − e σ − zσ ν dσσ − ν , z ∈ C , < ν < F ν ( z ) and M ν ( z ) in terms of known functionsare expected for some particular values of ν as shown and recalled byMainardi in the first 1990’s in a series of papers [29, 30, 31, 32], that is(14) M / ( z ) = 1 √ π e − z / , (15) M / ( z ) = 3 / Ai( z/ / ) , . Liemert and Klenie [21] have added the following expression for ν = 2 / M / ( z ) = 3 − / (cid:2) / z Ai (cid:0) z / / (cid:1) − (cid:48) (cid:0) z / / (cid:1)(cid:3) e − z / , where Ai and Ai (cid:48) denote the Airy function and its first derivative.Furthermore they have suggested in the positive real field IR + thefollowing remarkably integral representation(17) M ν ( x ) = 1 π x ν/ (1 − ν ) − ν (cid:90) π C ν ( φ ) exp ( − C ν ( φ )) x / (1 − ν ) dφ, where(18) C ν ( φ ) = sin(1 − ν )sin φ (cid:18) sin νφ sin φ (cid:19) ν/ (1 − ν ) , corresponding to equation (7) of the article written by Saa and Ve-negeroles [42] .Let us point out the asymptotic behaviour of the function M ν ( x ) as x → + ∞ . Choosing as a variable x/ν rather than x , the computationof the asymptotic representation by the saddle-point approximationcarried out by Mainardi and Tomirotti yields, see [39] and [35],(19) M ν ( x/ν ) ∼ a ( ν ) x ( ν − / / (1 − ν ) exp (cid:104) − b ( ν ) x / (1 − ν ) (cid:105) , where(20) a ( ν ) = 1 (cid:112) π (1 − ν ) > , b ( ν ) = 1 − νν > . The above evaluation is consistent with the first term in the originalasymptotic expansion by Wright in [49, 50] after having used the defini-tion of . M -Wright functionNow we find it convenient to show the plots of the M -Wright functionson a space symmetric interval of IR in Figs 1, 2, corresponding to thecases 0 ≤ ν ≤ / / ≤ ν ≤
1, respectively. We recognize the non-negativity of the M -Wright function on IR for 1 / ≤ ν ≤ FRANCESCO MAINARDI AND ARMANDO CONSIGLIO Figure 1.
Plots of the M -Wright function as a functionof the x variable, for 0 ≤ ν ≤ / Figure 2.
Plots of the M -Wright function as a functionof the x variable, for 1 / ≤ ν ≤ HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS The Wright functions of the second kind and thetime-fractional diffusion wave equation
As we will see the Wright functions of the second kind are relevant in theanalysis of the Time-Fractional Diffusion-Wave Equation (TFDWE).For this purpose we introduce now the TFDWE as a generalizationof the standard diffusion equation and we see how the two Mainardiauxiliary functions come into play.The TFDWE is so obtained from the standard diffusion equation (or theD’Alembert wave equation) by replacing the first-order (or the second-order) time derivative by a fractional derivative (of order 0 < β ≤
2) inthe Caputo sense, obtaining the following Fractional PDE:(21) ∂ β u∂t β = D ∂ u∂x < β ≤ , D > , where D is a positive constant whose dimensions are L T − β and u = u ( x, t ; β ) is the field variable, which is assumed again to be a causalfunction of time. The Caputo fractional derivative is recalled in theAppendix B so that in explicit form the TFDWE (21) splits in thefollowing integro-differential equations:(22) 1Γ(1 − β ) (cid:90) t ( t − τ ) − β (cid:18) ∂u∂τ (cid:19) dτ = D ∂ u∂x , < β ≤ − β ) (cid:90) t ( t − τ ) − β (cid:18) ∂ u∂τ (cid:19) dτ = D ∂ u∂x , < β ≤ . In view of our analysis we find convenient to put:(24) ν = β , < ν ≤ . We can then formulate the basic problems for the Time FractionalDiffusion-Wave Equation using a correspondence with the two problemsfor the standard diffusion equation.Denoting by f ( x ) and g ( t ) two given, sufficiently well-behaved functions,we define:a) Cauchy problem(25) (cid:40) u ( x, + ; ν ) = f ( x ) , −∞ < x < + ∞ ; u ( ±∞ , t ; ν ) = 0 , t > (cid:40) u ( x, + ; ν ) = 0 , ≤ x < + ∞ ; u (0 + , t ; ν ) = g ( t ) , u (+ ∞ , t ; ν ) = 0 , t > / < ν ≤ < β ≤ FRANCESCO MAINARDI AND ARMANDO CONSIGLIO initial value of the first time derivative of the field variable u t ( x, + ; ν ),since in this case Eq. (21) turns out to be akin to the wave equationand consequently two linear independent solutions are to be determined.However, to ensure the continuous dependence of the solutions to ourbasic problems on the parameter ν in the transition from ν = (1 / − to ν = (1 / + , we agree to assume u t ( x, + ; ν ) = 0.For the Cauchy and Signalling problems, following the approachesby Mainardi, see e.g. [29] and related papers, we introduce now theGreen functions G c ( x, t ; ν ) and G s ( x, t ; ν ) that for both problems can bedetermined by the LT technique, so extending the results known fromthe ordinary diffusion equation. We recall that the Green functions arealso referred to as the fundamental solutions, corresponding respectivelyto f ( x ) = δ ( x ) and g ( t ) = δ ( t ) with δ ( · ) is the Dirac delta generalizedfunctionThe expressions for the Laplace Transforms of the two Green’s functionsare:(27) (cid:101) G c ( x, s ; ν ) = 12 √ Ds − ν e ( −| x | / √ D ) s ν and(28) (cid:101) G s ( x, s ; ν ) = e − ( x/ √ D ) s ν Now we can easily recognize the following relation:(29) dds (cid:101) G s = − ν x (cid:101) G c , x > reciprocityrelation for x > t > < ν < νx G c ( x, t ; ν ) = t G s ( x, t ; ν ) = F ν ( z ) = νzM ν ( z ) , z = x √ Dt ν where z is the similarity variable and F ν ( z ) and M ν ( z ) are the Mainardiauxilary functions introduced in the previous section. Indeed Eq. (30)is the generalization of Eq. (A.8) that we have seen for the standarddiffusion equation due to the introduction of the time fractional deriva-tive of order ν Then, the two Green functions of the Cauchy and Signalling problemsturn out to be expressed in terms of the two auxiliary functions asfollows.For the Cauchy problem we have(31) G C ( x, t ; ν ) = t − ν √ D M ν (cid:18) | x |√ Dt ν (cid:19) , −∞ < x < + ∞ , t ≥ , HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS that generalizes Eq. (A.5).For the Signalling problem we have:(32) G S ( x, t ; ν ) = νxt − ν − √ D M ν (cid:18) x √ Dt ν (cid:19) , x ≥ , t ≥ , that generalizes Eq. (A.7).4.1. Complements to the time-fractional diffusion-wave equa-tions.
The boundary value problems dealt previously can be consideredwith a source data function f x ) and g ( t ) different from the Dirac gen-eralized functions, in particular with box-type functions as it has beencarried out recently by us, see [8].The TFDWE can be generalized in 2D and 3D space dimensions.so consequently the Wright functions play again a fundamental role.However, we prefer to refer the interested reader to the literature, inparticular to the papers by Luchko and collaborators [22, 23, 24, 25],[27, 28], [1], by Hanyga [18] and to the recent analysis by Kemppainen[19]. All of them are originated in some way from the seminal paperby Schneider & Wyss [43]. In some of these papers the authors haveconsidered also fractional differentiation both in time and in space,so that they have generalized to more than one dimension the formeranalysis by Mainardi, Luchko & Pagnini [37] on the space-time fractionaldiffusion-wave equations.5. The M − Wright functions in probability theory and thestable distributions
We recognize that the Wright M -function with support in IR + canbe interpreted as probability density function ( pdf ) because it is nonnegative and also it satisfies the normalization condition:(33) (cid:90) ∞ M ν ( x ) dx = 1 . We now provide more details on these densities in the framework of thetheory of probability.Fundamental quantities about the Wright M − function are the absolutemoments of order δ > − R + , that are finite and turn out to be:(34) (cid:90) ∞ x δ M ν ( x ) dx = Γ( δ + 1)Γ( νδ + 1) , δ > − , ≤ ν < . The result is based on the integral representation of the M − Wrightfunction: AND ARMANDO CONSIGLIO (cid:90) ∞ x δ M ν ( x ) dx = (cid:90) ∞ x δ (cid:34) πi (cid:90) Ha − e σ − xσ ν dσσ − ν (cid:35) dx = 12 πi (cid:90) Ha − e σ (cid:34)(cid:90) ∞ e − xσ ν x δ dx (cid:35) dσσ − ν = Γ( δ + 1)2 πi (cid:90) Ha − e σ σ νδ +1 dσ = Γ( δ + 1)Γ( νδ + 1)(35)The exchange between two integrals and the following identity con-tributed to the final result for Eq. (35):(36) (cid:90) ∞ e − xσ ν x δ dx = Γ( δ + 1)( σ ν ) δ +1 . In particular, for δ = n ∈ IN, the above formula provides the momentsof integer order. Indeed recalling the Mittag-Leffler function introducedin Eq. (5) with α = ν and β = 1:(37) E ν ( z ) := ∞ (cid:88) n =0 z n Γ( νn + 1) , ν > , z ∈ C , the moments of integer order can also be computed from the Laplacetransform pair(38) M ν ( x ) ÷ E ν ( s ) , proved in the Appendix F of [35] as follows:(39) (cid:90) + ∞ x n M ν ( x ) dx = lim s → ( − n d n ds n E ν ( − s ) = Γ( n + 1)Γ( νn + 1) . The auxiliary functions versus extremal stable densities.
We find it worthwhile to recall the relations between the Mainardiauxilary functions and the extremal L´evy stable densities as proven inthe 1997 paper by Mainardi and Tomirotti [40]. For readers’ conveniencewe refer to Appendix C for an essential account of the general L´evy stabledistributions in probability. Indeed, from a comparison between theseries expansions of stable densities in (C.8)-(C.9) and of the auxiliaryfunctions in Eqs. (9) - (10), we recognize that the auxiliary functionsare related to the extremal stable densities as follows(40) L − αα ( x ) = 1 x F α ( x − α ) = αx α +1 M α ( x − α ) , < α < , x ≥ , (41) L α − α ( x ) = 1 x F /α ( x ) = 1 α M /α ( x ) , < α ≤ , −∞ < x < + ∞ . In the above equations, for α = 1, the skewness parameter turns out tobe θ = −
1, so we get the singular limit(42) L − ( x ) = M ( x ) = δ ( x − . HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS Hereafter we show the plots the extremal stable densities according totheir expressions in terms of the M -Wright functions, see Eq. (40).,Eq. (41) for α = 1 / α = 3 /
2, respectively. We recognize that
Figure 3.
Plot of the unilateral extremal stable pdf for α = 1 / Figure 4.
Plot of the bilateral extremal stable pdf for α = 3 / The symmetric M − Wright function.
We easily recognizethat extending the function M ν ( x ) in a symmetric way to all of IR (thatis putting x = | x | ) and dividing by 2 we have a symmetric pdf withsupport in all of IR.As the parameter ν changes between 0 and 1, the pdf goes from theLaplace pdf to two half discrete delta pdf s passing for ν = 1 / pdf .To develop a visual intuition, also in view of the subsequent applications,we show the plots of the symmetric M − Wright function on the realaxis at t = 1 for some rational values of the parameter ν ∈ [0 , AND ARMANDO CONSIGLIO Figure 5.
Plot of the symmetric M − Wright function M ν ( | x | ) for 0 ≤ ν ≤ /
2. Note that the M − Wrightfunction becomes a Gaussian density for ν = 1 / Figure 6.
Plot of the symmetric M − Wright type func-tion M ν ( | x | ) | for 1 / ≤ ν ≤
1. Note that the M Wrightfunction becomes a a sum of two delta functions centeredin x = ± ν = 1.The readers are invited to look the YouTube video by Consiglio whosetitle is “Simulation of the M − Wright function”, in which the authorshows the evolution of this function as the parameter ν changes between0 and 0.85 in a finite interval of IR centered in x = 0.Finally we compute the characteristic function for the symmetric HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS M − Wright pdf . We get for M ν ( | x |F (cid:104) M ν ( | x | ) (cid:105) := 12 (cid:90) + ∞−∞ e + iκx M ν ( | x | ) dx = (cid:90) ∞ cos ( κx ) M ν ( x ) dx = E ν ( − κ )(43)Eq. 43 is obtained by developing in series the cosine function andusing Eq. 35. In particular: (cid:90) ∞ cos ( κx ) M ν ( x ) dx = ∞ (cid:88) n =0 ( − n κ n (2 n )! (cid:90) ∞ x n M ν ( x ) dx = ∞ (cid:88) n =0 ( − n κ n Γ(2 νn + 1) = E ν ( − κ )(44)5.3. The Wright M -function in two variables. In view of time-fractional diffusion processes related to time-fractional diffusion equa-tions it is worthwhile to introduce the function in two variables(45) M ν ( x, t ) := t − ν M ν ( xt − ν ) , < ν < , x, t ∈ IR + , which defines a spatial probability density in x evolving in time t withself-similarity exponent H = ν . Of course for x ∈ IR we have to considerthe symmetric version of the M -Wright function.Hereafter we provide a list of the main properties of this function,which can be derived from the Laplace and Fourier transforms for thecorresponding Wright M -function in one variable.From Eqs. (39) and (43) we derive the Laplace transform of M ν ( x, t )with respect to t ∈ IR + ,(46) L { M ν ( x, t ); t → s } = s ν − e − xs ν . From Eq. (18) we derive the Laplace transform of M ν ( x, t ) with respectto x ∈ IR + ,(47) L { M ν ( x, t ); x → s } = E ν ( − st ν ) . From Eq. (55) we derive the Fourier transform of M ν ( | x | , t ) with respectto x ∈ IR,(48)
F { M ν ( | x | , t ); x → κ } = 2 E ν (cid:0) − κ t ν (cid:1) . Using the Mellin transforms, Mainardi et al. [38] derived the followinginteresting integral formula of composition,(49) M ν ( x, t ) = (cid:90) ∞ M λ ( x, τ ) M µ ( τ, t ) dτ , ν = λµ . Special cases of the Wright M -function are simply derived for ν = 1 / ν = 1 / AND ARMANDO CONSIGLIO Eqs. (28)-(29). We devote particular attention to the case ν = 1 / M / ( | x | , t ) = 12 √ πt / e − x / (4 t ) . For the limiting case ν = 1 we obtain(51) M ( | x | , t ) = 12 [ δ ( x − t ) + δ ( x + t )] . The four sisters
In this section we show how some Wright functions of the second kindcan provide an interesting generalization of the three sisters discussed inAppendix A. The starting point is a (not well- known) paper publishedin 1970 by Stankovic [45], where (in our notation) the following Laplacetransform pair is proved rigorously:(52) t µ − W − ν,µ ( x, t ) ÷ s − µ e − xs ν < ν < , µ ≥ , where x and t are positive. We note that the Stankovic formula canbe derived in a formal way by developing the exponential function inpositive power of s and inverting term by term as described in theAppendix F of the book by Mainardi [35].We recognize that the Laplace Transforms of the Three Sisters functions (cid:101) φ ( x, s ), (cid:101) ψ ( x, s ) and (cid:101) χ ( x, s ) are particular cases of the Eq. (52) for ν = 1 /
2, that is of(53) t µ − W − / ,µ ( x, t ) ÷ s − µ e − x √ s , according to the following scheme:- (cid:101) φ ( x, s ) with µ = 1,- (cid:101) ψ ( x, s ) with µ = 0,- (cid:101) χ ( x, s ) with µ = 1 / ν is no longer restricted to ν = 1 / Four Sisters functions as follows µ = 0 , e − xs ν ÷ t − W − ν, ( − xt − ν ) ,µ = 1 − ν, e − xs ν s − ν ÷ t − ν W − ν, − ν ( − xt − ν ) ,µ = ν, e − xs ν s ν ÷ t ν − W − ν,ν ( − xt − ν ) ,µ = 1 , e − xs ν s ÷ W − ν, ( − xt − ν ) . (54)Here we show some plots of these functions, both in the t and in the x domain for some values of ν ( ν = 1 / , / , / ν = 1 / HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS Figure 7.
The four sisters functions in linear scale with ν = 1 /
4; top: versus t ( x = 1), bottom: versus x ( t = 1) Figure 8.
The three sisters functions in linear scale with ν = 1 /
2; top: versus t ( x = 1), bottom: versus x ( t = 1) AND ARMANDO CONSIGLIO Figure 9.
The four sisters functions in linear scale with ν = 3 /
4; top: versus t ( x = 1), bottom: versus x ( t = 1)7. Conclusions
In our survey on the Wright functions we have distinguished twokinds, pointing out the particular class of the second kind. Indeedthese functions have been shown to play key roles in several processesgoverned by non Gaussian processes, including sub-diffusion, transitionto wave propagation, L´evy stable distributions. Furthermore, we havedevoted our attention to four functions of this class that we agree tocalled the Four Sisters functions . All these items justify the relevanceof the Wright functions of the second kind in Mathematical Physics.
Acknowledgments
The research activity of both the authors has been carried out inthe framework of the activities of the National Group of MathematicalPhysics (GNFM, INdAM).
HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS Appendix A: The standard diffusion equation and thethree sisters
In this Appendix let us recall the Diffusion Equation in the one-dimensional case ∂u∂t = D ∂ u∂x , ( A. D > L T − and x , t denote the space and time coordinates,respectively.Two basic problems for Eq. (7) are the Cauchy and
Signalling onesintroduced hereafter In these problems some initial values and bound-ary conditions are set; specify the values attained by the field variableand/or by some of its derivatives on the boundary of the space-timedomain is an essential step to guarantee the existence, the uniquenessand the determination of a solution of physical interest to the problem,not only for the Diffusion Equation.Two data functions f ( x ) and g ( t ) are then introduced to write for-mally these conditions; some regularities are required to be satisfied by f ( x ) and g ( t ), and in particular f ( x ) must admit the Fourier transformor the Fourier series expansion if the support is finite, while h ( t ) mustadmit the Laplace Transform.We also require without loss of generality that the field variable u ( x, t )is vanishing for t < x in the spatial domain. Given thesepremises, we can specify the two aforementioned problems.In the Cauchy problem the medium is supposed to be unlimited ( −∞
0, as follows φ ( a, t ) := erfc (cid:16) a √ t (cid:17) ÷ e − as / s := (cid:101) φ ( a, s ) , ( A. ψ ( a, t ) := a √ π t − / e − a / (4 t ) ÷ e − as / := (cid:101) ψ ( a, s ) , ( A. χ ( a, t ) := 1 √ π t − / e − a / (4 t ) ÷ e − as / s / := (cid:101) χ ( a, s ) ( A. HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS where the sign ÷ is used for the juxtaposition of a function with itsLaplace transform. We easily note that Eq. (7) is related to the Step-Response problem, Eq. (7) is related to the Signalling problem andEq. (7) is related to the Cauchy problem. Following the lecture notesby Mainardi [34] we agree to call the above functions the three sistersfunctions for their role in the standard diffusion equation. They will bediscussed with details hereafter.Everything that we have said above will be found again as a special caseof the Time Fractional Diffusion Equation where the time derivative ofthe first order is replaced by a suitable time derivative of non-integerorder.It is easy to demonstrate that each of them can be expressed as afunction of one of the 2 others three sisters (table 1). (cid:101) φ (cid:101) ψ (cid:101) χ (cid:101) φ e − a √ s s (cid:101) ψs − s ∂ (cid:101) χ∂a (cid:101) ψ s (cid:101) φ e − a √ s − ∂ (cid:101) χ∂a (cid:101) χ − ∂ (cid:101) φ∂a − a ∂ (cid:101) ψ∂s e − a √ s √ s Table 1.
Relations among the three sisters in theLaplace domain.The three sisters in the t domain may be all directly calculated bymaking use of the Bromwich formula taking account of the contributionof the branch cut of √ s and of the pole of 1 /s . Wee obtain: (cid:101) φ ( a, s ) ÷ φ ( a, t ) = 1 − π (cid:90) ∞ e − rt sin( a √ r ) d rr , ( A. a ) (cid:101) ψ ( a, s ) ÷ ψ ( a, t ) = 1 π (cid:90) ∞ e − rt sin( a √ r ) d r , ( A. a ) (cid:101) χ ( a, s ) ÷ χ ( a, t ) = 1 π (cid:90) ∞ e − rt cos( a √ r ) d r √ r . ( A. a )Then, through the substitution ρ = √ r , we arrive at the Gaussianintegral and, consequently, we find the previous explicit expressions ofthe three sisters , that is: φ ( a, t ) = erfc( a √ t ) = 1 − √ π (cid:90) a/ √ t e − u d u , ( A. ψ ( a, t ) = a √ π t − / e − a / t , ( A. AND ARMANDO CONSIGLIO χ ( a, t ) = 1 √ π t − / e − a / t , ( A. three sisters in t domain by usingthe relations among the three sisters in the Laplace domain listed intable 1. But in this case one of the three sisters in t domain must bealready known. Assuming to know φ ( a, t ) from Eq. (A.11), we get:- ψ ( a, t ) from (cid:101) ψ ( a, s ) = s (cid:101) φ ( a, s ). Indeed, noting s (cid:101) φ ( a, s ) ÷ ∂∂t φ ( a, t )since φ ( a, + ) = 0 we can obtain (A.12), namely ψ ( a, t ) = a √ π t − / e − a / t ;- χ ( a, t ) from (cid:101) χ ( a, s ) = − ∂∂a (cid:101) φ ( a, s ) where a is seen as a parameter,.Indeed it immediately follows (A.13), namely χ ( a, t ) = − ∂∂a φ ( a, t ) = 1 √ π t − / e − a / t . For more details we refer the reader again to [34].
Appendix B: Essentials of Fractional Calculus
Fractional calculus is the field of mathematical analysis which dealswith the investigation and applications of integrals and derivatives ofarbitrary order. The term fractional is a misnomer, but it is retainedfor historical reasons, following the prevailing use.This appendix is based on the 1997 surveys by Gorenflo and Mainardi[16] and by Mainardi [33]. For more details on the classical treatment offractional calculus the reader is referred to the nice and rigorous bookby Diethelm [9] published in 2010 by Springer in the series LectureNotes in Mathematics.According to the Riemann-Liouville approach to fractional calculus, thenotion of fractional integral of order α ( α >
0) is a natural consequenceof the well known formula (usually attributed to Cauchy), that reducesthe calculation of the n − fold primitive of a function f ( t ) to a singleintegral of convolution type. In our notation the Cauchy formula reads J n f ( t ) := f n ( t ) = 1( n − (cid:90) t ( t − τ ) n − f ( τ ) dτ , t > , n ∈ IN , ( B. f n ( t ) vanishes at t = 0 with its derivatives of order 1 , , . . . , n − . HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS For convention we require that f ( t ) and henceforth f n ( t ) be a causal function, i.e. identically vanishing for t < . In a natural way one is led to extend the above formula from pos-itive integer values of the index to any positive real values by us-ing the Gamma function. Indeed, noting that ( n − n ) , and introducing the arbitrary positive real number α , one defines the Fractional Integral of order α > J α f ( t ) := 1Γ( α ) (cid:90) t ( t − τ ) α − f ( τ ) dτ , t > , α ∈ IR + , ( B. + is the set of positive real numbers. For complementationwe define J := I (Identity operator), i.e. we mean J f ( t ) = f ( t ) . Furthermore, by J α f (0 + ) we mean the limit (if it exists) of J α f ( t ) for t → + ; this limit may be infinite.We note the semigroup property J α J β = J α + β , α , β ≥ , whichimplies the commutative property J β J α = J α J β , and the effect of ouroperators J α on the power functions J α t γ = Γ( γ + 1)Γ( γ + 1 + α ) t γ + α , α ≥ , γ > − , t > . ( B. L { f ( t ) } := (cid:82) ∞ e − st f ( t ) dt = (cid:101) f ( s ) , s ∈ C , and using the sign ÷ to denote a Laplace transform pair, i.e. f ( t ) ÷ (cid:101) f ( s ) , we note the following rule for the Laplace transform ofthe fractional integral, J α f ( t ) ÷ (cid:101) f ( s ) s α , α ≥ , ( B. n -fold repeated integral.After the notion of fractional integral, that of fractional derivative oforder α ( α >
0) becomes a natural requirement and one is attempted tosubstitute α with − α in the above formulas. However, this generalizationneeds some care in order to guarantee the convergence of the integralsand preserve the well known properties of the ordinary derivative ofinteger order.Denoting by D n with n ∈ IN , the operator of the derivative of order n , we first note that D n J n = I , J n D n (cid:54) = I , n ∈ IN , i.e. D n is left-inverse (and not right-inverse) to the corresponding integral operator J n . In fact we easily recognize from (B.1) that J n D n f ( t ) = f ( t ) − n − (cid:88) k =0 f ( k ) (0 + ) t k k ! , t > . ( B. AND ARMANDO CONSIGLIO As a consequence we expect that D α is defined as left-inverse to J α .For this purpose, introducing the positive integer m such that m − < α ≤ m , one defines the Fractional Derivative of order α > D α f ( t ) := D m J m − α f ( t ) , i.e. D α f ( t ) := d m dt m (cid:20) m − α ) (cid:90) t f ( τ )( t − τ ) α +1 − m dτ (cid:21) , m − < α < m,d m dt m f ( t ) , α = m . ( B. D = J = I , then we easily recognizethat D α J α = I , α ≥ , and D α t γ = Γ( γ + 1)Γ( γ + 1 − α ) t γ − α , α ≥ , γ > − , t > . ( B. D α f is not zerofor the constant function f ( t ) ≡ α (cid:54)∈ IN . In fact, (B.7) with γ = 0teaches us that D α t − α Γ(1 − α ) , α ≥ , t > . ( B. ≡ α ∈ IN, due to the poles of the gammafunction in the points 0 , − , − , . . . . We now observe that an alternativedefinition of fractional derivative was introduced by Caputo in 1967 [3]in a geophysical journal and in 1969 [4] in a book in Italian. Then theCaputo definition was adopted in 1971 by Caputo and Mainardi [5, 6]in the framework of the theory of Linear Viscoelasticity . Nowadaysit is usually referred to as the
Caputo fractional derivative and reads D α ∗ f ( t ) := J m − α D m f ( t ) with m − < α ≤ m , m ∈ IN , i.e. D α ∗ f ( t ) := m − α ) (cid:90) t f ( m ) ( τ )( t − τ ) α +1 − m dτ , m − < α < m ,d m dt m f ( t ) , α = m . ( B. Caputo fractional derivative to distinguishit from the standard Riemann-Liouville fractional derivative (B.6).The Caputo definition (B.9) is of course more restrictive than the
HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS Riemann-Liouville definition (B.6), in that requires the absolute inte-grability of the derivative of order m . Whenever we use the operator D α ∗ we (tacitly) assume that this condition is met. We easily recognizethat in general D α f ( t ) := D m J m − α f ( t ) (cid:54) = J m − α D m f ( t ) := D α ∗ f ( t ) , ( B. f ( t ) along with its first m − t = 0 + . In fact, assuming that the passage of the m -derivative underthe integral is legitimate, one recognizes that, for m − < α < m and t > , D α f ( t ) = D α ∗ f ( t ) + m − (cid:88) k =0 t k − α Γ( k − α + 1) f ( k ) (0 + ) , ( B. D α (cid:32) f ( t ) − m − (cid:88) k =0 t k k ! f ( k ) (0 + ) (cid:33) = D α ∗ f ( t ) . ( B. m − t = 0 + from f ( t ) means a sort of regularization of the Riemann-Liouvillefractional derivative. In particular for 0 < α < D α (cid:0) f ( t ) − f (0 + ) (cid:1) = D α ∗ f ( t ) . According to the Caputo definition, the relevant property for which thefractional derivative of a constant is still zero can be easily recognized, i.e. D α ∗ ≡ , α > . ( B. D α t α − ≡ , α > , t > . From above we thus recognize thefollowing statements about functions which for t > α , with m − < α ≤ m , m ∈ IN ,D α f ( t ) = D α g ( t ) ⇐⇒ f ( t ) = g ( t ) + m (cid:88) j =1 c j t α − j , ( B. D α ∗ f ( t ) = D α ∗ g ( t ) ⇐⇒ f ( t ) = g ( t ) + m (cid:88) j =1 c j t m − j . ( B. c j are arbitrary constants. AND ARMANDO CONSIGLIO For the two definitions we also note a difference with respect to the formal limit as α → ( m − + . From (B.6) and (B.9) we obtainrespectively, α → ( m − + = ⇒ D α f ( t ) → D m J f ( t ) = D m − f ( t ) ; ( B. α → ( m − + = ⇒ D α ∗ f ( t ) → J D m f ( t ) = D m − f ( t ) − f ( m − (0 + ) . ( B. Laplace transform of the two fractional derivatives.For the standard fractional derivative D α the Laplace transform, as-sumed to exist, requires the knowledge of the (bounded) initial valuesof the fractional integral J m − α and of its integer derivatives of order k = 1 , , . . . , m − . The corresponding rule reads, in our notation, D α f ( t ) ÷ s α (cid:101) f ( s ) − m − (cid:88) k =0 D k J ( m − α ) f (0 + ) s m − − k , m − < α ≤ m . ( B. Caputo fractional derivative appears more suitable to be treatedby the Laplace transform technique in that it requires the knowledge ofthe (bounded) initial values of the function and of its integer derivativesof order k = 1 , , . . . , m − , in analogy with the case when α = m . Infact, by using (B.4) and noting that J α D α ∗ f ( t ) = J α J m − α D m f ( t ) = J m D m f ( t ) = f ( t ) − m − (cid:88) k =0 f ( k ) (0 + ) t k k ! . ( B. D α ∗ f ( t ) ÷ s α (cid:101) f ( s ) − m − (cid:88) k =0 f ( k ) (0 + ) s α − − k , m − < α ≤ m . ( B. α = m . In particular Gorenflo and Mainardi have pointed out the major utility ofthe Caputo fractional derivative in the treatment of differential equationsof fractional order for physical applications . In fact, in physical problems,the initial conditions are usually expressed in terms of a given numberof bounded values assumed by the field variable and its derivatives ofinteger order, no matter if the governing evolution equation may bea generic integro-differential equation and therefore, in particular, afractional differential equation.
HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS Appendix C: The L´evy stable distributions
We now introduce the so-called
L´evy stable distributions .. The termstable has been assigned by the French mathematician Paul L´evy, who,in the tuenties of the last century, started a systematic research inorder to generalize the celebrated
Central Limit Theorem to probabilitydistributions with infinite variance. For stable distributions we canassume the following
Definition : If two independent real randomvariables with the same shape or type of distribution are combinedlinearly and the distribution of the resulting random variable has thesame shape, the common distribution (or its type, more precisely) issaid to be stable .The restrictive condition of stability enabled L´evy (and then otherauthors) to derive the canonic form for the characteristic function ofthe densities of these distributions. Here we follow the parameterizationby Feller [11, 12] revisited by Gorenflo & Mainardi in [17], see also[37]. Denoting by L θα ( x ) a generic stable density in IR, where α is the index of stability and and θ the asymmetry parameter, improperly called skewness , its characteristic function reads: L θα ( x ) ÷ (cid:98) L θα ( κ ) = exp (cid:2) − ψ θα ( κ ) (cid:3) , ψ θα ( κ ) = | κ | α e i (sign κ ) θπ/ , ( C. < α ≤ , | θ | ≤ min { α, − α } . We note that the allowed region for the parameters α and θ turnsout to be a diamond in the plane { α, θ } with vertices in the points(0 , , (1 , , (1 , − , (2 , Feller-Takayasu diamond ,see Figure 10. For values of θ on the border of the diamond (that is θ = ± α if 0 < α <
1, and θ = ± (2 − α ) if 1 < α <
2) we obtain theso-called extremal stable densities . Figure 10.
The Feller-Takayasu diamond for L´evy sta-ble densities. AND ARMANDO CONSIGLIO We note the symmetry relation L θα ( − x ) = L − θα ( x ), so that a stabledensity with θ = 0 is symmetric.Stable distributions have noteworthy properties of which the interestedreader can be informed from the relevant existing literature. Here-afterwe recall some peculiar Properties :- The class of stable distributions possesses its own domain of attraction ,see e.g. [12].-
Any stable density is unimodal and indeed bell-shaped , i.e. its n -thderivative has exactly n zeros in IR, see Gawronski [13], Simon [44] andKwa´snicki [20].- The stable distributions are self-similar and infinitely divisible .These properties derive from the canonic form (C.1) through the scalingproperty of the Fourier transform.
Self-similarity means L θα ( x, t ) ÷ exp (cid:2) − tψ θα ( κ ) (cid:3) ⇐⇒ L θα ( x, t ) = t − /α L θα ( x/t /α )] , ( C. t is a positive parameter. If t is time, then L θα ( x, t ) is a spatialdensity evolving on time with self-similarity. Infinite divisibility means that for every positive integer n , the character-istic function can be expressed as the n th power of some characteristicfunction, so that any stable distribution can be expressed as the n -foldconvolution of a stable distribution of the same type. Indeed, taking in(C.1) θ = 0, without loss of generality, we havee − t | κ | α = (cid:2) e − ( t/n ) | κ | α (cid:3) n ⇐⇒ L α ( x, t ) = (cid:2) L α ( x, t/n ) (cid:3) ∗ n , ( C. (cid:2) L α ( x, t/n ) (cid:3) ∗ n := L α ( x, t/n ) ∗ L α ( x, t/n ) ∗ · · · ∗ L α ( x, t/n )is the multiple Fourier convolution in IR with n identical terms.Only for a few particular cases, the inversion of the Fourier transformin (C.1) can be carried out using standard tables, and well-knownprobability distributions are obtained.For α = 2 (so θ = 0), we recover the Gaussian pdf , that turns out to bethe only stable density with finite variance, and more generally withfinite moments of any order δ ≥
0. In fact L ( x ) = 12 √ π e − x / . ( C. δ ∈ [ − , α ) as we will later show.For α = 1 and | θ | <
1, we get L θ ( x ) = 1 π cos( θπ/ x + sin( θπ/ + [cos( θπ/ , ( C. HE WRIGHT FUNCTIONS OF THE SECOND KINDIN MATHEMATICAL PHYSICS which for θ = 0 includes the Cauchy-Lorentz pdf . L ( x ) = 1 π
11 + x . ( C. θ = ± α = 1 we obtain the singular Diracpdf ’s L ± ( x ) = δ ( x ± . ( C. x > x < L θα ( x ) ( x >
0) turn out to be;for 0 < α < , | θ | ≤ α : L θα ( x ) = 1 π x ∞ (cid:88) n =1 ( − x − α ) n Γ(1 + nα ) n ! sin (cid:104) nπ θ − α ) (cid:105) ; ( C, < α ≤ , | θ | ≤ − α : L θα ( x ) = 1 π x ∞ (cid:88) n =1 ( − x ) n Γ(1 + n/α ) n ! sin (cid:104) nπ α ( θ − α ) (cid:105) . ( C. theextremal stable densities for < α < are unilateral , precisely vanishingfor x > θ = α , vanishing for x < θ = − α . In particular theunilateral extremal densities L − αα ( x ) with 0 < α < + and Laplace transform exp( − s α ). For α = 1 / L´evy-Smirnov pdf : L − / / ( x ) = x − / √ π e − / (4 x ) , x ≥ . ( C. pdf ’s with 1 < α ≤ < α < L θα ( x ) = (cid:40) (1 /x ) Φ ( x − α ) for x > , (1 / | x | ) Φ ( | x | − α ) for x < , ( C. ( z ) and Φ ( z ) are distinct entire functions.The case α = 1 ( | θ | <
1) must be considered in the limit for α → /x and x , respectively, with a finite radius of convergence. Thecorresponding stable pdf ’s are no longer represented by entire functions,as can be noted directly from their explicit expressions (C.5)-(C.6).We omit to provide the asymptotic representations of the stable densitiesreferring the interested reader to Mainardi et al (2001) [37]. However,based on asymptotic representations, we can state as follows; for 0 < AND ARMANDO CONSIGLIO α < pdf ’s exhibit fat tails in such a way that their absolutemoment of order δ is finite only if − < δ < α . More precisely, onecan show that for non-Gaussian, not extremal, stable densities theasymptotic decay of the tails is L θα ( x ) = O (cid:0) | x | − ( α +1) (cid:1) , x → ±∞ . ( C. α (cid:54) = 1 this is valid only for one tail(as | x | → ∞ ), the other (as | x | → ∞ ) being of exponential order. For1 < α < pdf ’s are two-sided and exhibit an exponentialleft tail (as x → −∞ ) if θ = +(2 − α ) , or an exponential right tail(as x → + ∞ ) if θ = − (2 − α ) . Consequently, the Gaussian pdf isthe unique stable density with finite variance. Furthermore, when0 < α ≤
1, the first absolute moment is infinite so we should usethe median instead of the non-existent expected value in order tocharacterize the corresponding pdf .Let us also recall a relevant identity between stable densities with index α and 1 /α (a sort of reciprocity relation) pointed out in [12], that is,assuming x > x α +1 L θ /α ( x − α ) = L θ ∗ α ( x ) , / ≤ α ≤ , θ ∗ = α ( θ + 1) − . ( C. / ≤ α ≤ ≤ /α ≤
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E-mail address : [email protected] (Corresponding Author) Institut f¨ur Theoretische Physik und Astrophysik,, Universit¨atW¨urzburg, D-97074 W¨urzburg, Germany
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