Differentiation of the Wright functions with respect to parameters and other results
DDIFFERENTIATION OF THE WRIGHTFUNCTIONS WITH RESPECT TOPARAMETERS AND OTHER RESULTS
October 29, 2020
Alexander Apelblat
Department of Chemical Engineering, Ben Gurion University of the Negev,Beer Sheva, 84105, Israel, [email protected]
Francesco Mainardi
Department of Physics and Astronomy, University of Bologna, Via Irnerio 46,40126 Bologna, Italy, [email protected]
Abstract:
In this survey we discuss the derivatives of the Wright functions (ofthe first and the second kind) with respect to parameters. Di ff erentiation ofthese functions leads to infinite power series with coe ffi cient being quotientsof the digamma (psi) and gamma functions. Only in few cases it is possible toobtain the sums of these series in a closed form. Functional form of the powerseries resembles those derived for the Mittag-Le ffl er functions. If the Wrightfunctions are treated as the generalized Bessel functions, di ff erentiation oper-ations can be expressed in terms of the Bessel functions and their derivativeswith respect to the order. It is demonstrated that in many cases it is possi-ble to derive the explicit form of the Mittag-Le ffl er functions by performingsimple operations with the Laplace transforms of the Wright functions. TheLaplace transform pairs of the both kinds of the Wright functions are discussedfor particular values of the parameters. Some transform pairs serve to obtainfunctional limits by applying the shifted Dirac delta function. Keywords:
Derivatives with respect to parameters; Wright functions; Mittag-Le ffl er functions; Laplace transforms; infinite power series; quotients of digammaand gamma functions; functional limits. MSC Classifications: a r X i v : . [ m a t h . G M ] O c t Introduction
Partial di ff erential equations of fractional order are successively applied formodeling time and space di ff usion, stochastic processes, probability distribu-tions and other diverse natural phenomena. They are extremely important inphysical processes that can be described by using the fractional calculus. In themathematical literature, when solution of these fractional di ff erential equa-tions is desired, we frequently encounter introduced and named after him,the Wright functions. At beginning, at 1933 [1] and at 1940 [2], these func-tions were considered as a some kind of generalization of the Bessel functions,but today they play an significant independent role in the theory of specialfunctions. There are many investigations devoted to analytical properties andapplications of the Wright functions, but here are mentioned only two surveypapers where essential material on the subject is included [3,4].In this survey we discuss three quite di ff erent subjects which are associatedwith the Wright functions. In the first part, the Wright function W α,β ( t ) where t is the argument and α and β are the (non-negative) parameters, is di ff eren-tiated with respect to parameters and derived expressions are compared withsimilar formulas for the Mittag-Le ffl er functions. In a continuous e ff ort, af-ter investigating di ff erentiation of the Bessel functions and the Mittag-Le ffl erfunctions with respect to their parameters [5-8], this mathematical operationis extended here to the Wright functions. Functional behaviour of derivativeswith respect to the order is also presented in graphical form. The presentedplots were prepared by evaluation of sums of infinite series by using MATHE-MATICA program. Special attention is devoted to the cases when the Wrightfunctions can be reduced to the Bessel functions and expressed in a closedform.Firthermore we extend our analysis to the so called Wright function of thesecond kind that exhibit a negative first parameter: alpha = − − sigma , with σ ∈ (0 , F σ ( t ) and M σ ( t ), which wereintroduced in 199o’s by Mainardi, see for details [4],The successive part of this survey is dedicated to the Laplace transformpairs of the Wright functions. It is demonstrated how the Laplace transformsof the Wright functions are useful for obtaining explicit expressions for theMittag-Le ffl er functions.Finally, we discuss the functional limits which are associated with the Wrightand the Mittag-Le ffl er functions. These limits can be derived by applying thedelta sequence in the form of the shifted Dirac function. This delta sequence isdirectly related to the order of Bessel function and was introduced by Lambornin 1969 [9]. 2hroughout this paper all mathematical operations or manipulations withfunctions, series, integrals, integral representations and transforms are formaland it is assumed that arguments and parameters are real numbers. Therewill be no proofs of validity of derived results, though they are presumed to becorrect considering that in a part they were previously obtained independentlyby other methods. The Wright functions W α,β ( z ) are defined by the series representation as a func-tion of complex argument z and parameters α and β . W α,β ( z ) = ∞ (cid:88) k =0 z k k ! Γ ( α k + β ) . (1)They are entire functions of zin C for α > − β (here al-ways β ≥ α ≥
0, and of the second kindfor − < α <
0. This distinctions is justified for the di ff erence in the aymptoticsrepresentations in the complex domain and in the Laplace transforms for thereal positive argument. For our purposes we recall their Laplace transformsfor positive argument t , We have, by using the symbol ÷ to denote the juxtapo-sition of a function f ( t ) with its Laplace transform (cid:101) f ( s ),i) for the first kind, when α ≥ W α,β ( ± t ) ÷ s E λ,µ (cid:18) ± s (cid:19) ; (2)ii) for the second kind, when − < α < σ = − α so0 < σ < W − σ,β ( − t ) ÷ E σ,β + σ ( − s ) . (3)Above we have introduced the Mittag-Le ffl er function in two parameters α > β ∈ C defined as its convergent series for all z ∈ C E α,β ( z ) := ∞ (cid:88) k =0 z k Γ ( αk + β ) . (4)For more details on the special functions of the Mittag-le ffl er type we refer theinterested readers to the treatise by Gorenflo et al [5] where in the forthcoming2-nd edition also the Wright functions are treated in some detail.3 Di ff erentiation of the Wright functionsof the first kind with respect to parameters We first compare compare the Wright functions of the first kind with the two-parameter Mittag-Le ffl er functions.Direct di ff erentiation of series with respect to the parameter α gives ∂W α,β ( t ) ∂α = − (cid:80) ∞ k = 1 (cid:16) ψ ( α k + β ) k ! Γ ( α k + β ) (cid:17) k t k = − (cid:80) ∞ k = 1 (cid:16) ψ ( α k + β )( k − Γ ( α k + β ) (cid:17) t k∂E α,β ( t ) ∂α = − (cid:80) ∞ k =1 (cid:16) ψ ( α k + β ) Γ ( α k + β ) (cid:17) k t k (5)and with respect to the parameter β ∂W α,β ( t ) ∂β = − (cid:80) ∞ k = 0 (cid:16) ψ ( α k + β ) k ! Γ ( α k + β ) (cid:17) t k∂E α,β ( t ) ∂β = − (cid:80) ∞ k = 0 (cid:16) ψ ( α k + β ) Γ ( α k + β ) (cid:17) t k (6)The second derivatives are ∂ W α,β ( t ) ∂α = (cid:80) ∞ k = 1 (cid:26) [ ψ ( α k +1)] − ψ (1) ( α k + β ) k ! Γ ( α k + β ) (cid:27) k t k∂ E α,β ( t ) ∂α = (cid:80) ∞ k = 1 (cid:26) [ ψ ( α k +1)] − ψ (1) ( α k + β ) Γ ( α k + β ) (cid:27) k t k (7)and ∂ W α,β ( t ) ∂β = (cid:80) ∞ k = 0 (cid:26) [ ψ ( α k + β )] − ψ (1) ( α k + β ) k ! Γ ( α k + β ) (cid:27) t k∂ E α,β ( t ) ∂β = (cid:80) ∞ k = 0 (cid:26) ψ ( α k + β )] − ψ (1) ( α k + β ) Γ ( α k + β ) (cid:27) t k (8)We note that for the Mittag-Le ffl er and the Wright functions we have thesame functional expressions, but in the case of the Wright functions factorialsappear. Contrary to the Mittag-Le ffl er functions [8], summation of these seriesby using MATHEMATICA gives only few results in a closed form in terms ofthe generalized hypergeometric functions ∂W α,β ( t ) ∂α (cid:12)(cid:12)(cid:12) α =1 ,β =0 = − (cid:80) ∞ k = 1 (cid:18) ψ ( k )[( k − (cid:19) t k = t F ( ; 1; t ) ∂W α,β ( t ) ∂α (cid:12)(cid:12)(cid:12) α =1 ,β =1 = − (cid:80) ∞ k = 1 (cid:16) ψ ( k +1)( k − k ! (cid:17) t k = t F ( ; 2; t ) (9)and ∂W α,β ( t ) ∂β (cid:12)(cid:12)(cid:12) α =1 ,β =0 = − (cid:80) ∞ k = 0 (cid:16) ψ ( k )( k − k ! (cid:17) t k = [ t F ( ; 1; t ) − t F ( ; 2; t ) ln t ] + √ tK (2 √ t ) ∂W α,β ( t ) ∂β (cid:12)(cid:12)(cid:12) α =1 ,β =1 = − (cid:80) ∞ k = 0 (cid:18) ψ ( k +1)( k !) (cid:19) t k = F ( ; 1; t ) (10)In the last case, α = β = 1, in the Brychkov compilation of infinite series [10],the sum is expressed in terms of the modified Bessel functions ∂W α,β ( t ) ∂β (cid:12)(cid:12)(cid:12) α = β =1 = − (cid:80) ∞ k =0 (cid:18) ψ ( k +1)( k !) (cid:19) t k = − ln t I (2 √ t ) − K (2 √ t ) (11)4sing MATHEMATICA program, values of derivatives with respect to pa-rameters α and β of the Wright functions of the first kind were calculated forthe argument 0 . ≤ t ≤ . ≤ α ≤ . ≤ β ≤ . α at di ff erent values if the argument t .As can be observed, in the 0 < α < α all curves tend to zero value. The absolute value of the minimum in-creases with increasing argument.Derivatives with respect to the parameters α and β when the argument t is constant, are presented in Figure 2. The functional form of curves with thechange of β values is similar to that observed previously in Figure 1.In order to compare the behaviour of derivatives with respect to α withthose with respect to β , the same conditions were imposed on t and β in Figures3 and 4 as are in Figures 1 and 2.As can be observed, the similarity of corresponding curves is evident, theonly di ff erence is that the absolute values of the minima are lower for deriva-tives with respect to parameter β than to for α . Fig. 1 : Derivatives of the Wright functions of the first kind with respect to pa-rameter α as a function of α for β = 1 and : t = 0 . : t = 1 . : t = 1 . : t = 1 . : t = 2 .
0. 5 ig.2 : Derivatives of the Wright functions of the first kind with respect to pa-rameter α as a function of α and β = 1 for t = 2 . Fig.3 : Derivatives of the Wright functions with respect to parameter β as afunction of α for β = 1 and : t = 0 . t = 1 . t = 1 . : t = 1 . t = 2 . og. 4 :Derivatives of the Wright functions with respect to parameter β as afunction of α and β = 1 for t = 2 . ff erentiation of some Wright functionswith respect to parameters versus Bessel functions Initially, the Wright functions (of the first kind) were treated as the generalizedBessel functions because for parameters α = 1 and β + 1 they become W ,β +1 ( − t ) = (cid:16) t (cid:17) β J β ( t ) W ,β +1 ( t ) = (cid:16) t (cid:17) β I β ( t ) (12)Di ff erentiation of the Wright functions in (12) with respect to parameter β gives ∂ (cid:18) W ,β +1 ( − t ) (cid:19) ∂β = (cid:16) t (cid:17) β (cid:20) ln (cid:16) t (cid:17) J β ( t ) + ∂J β ( t ) ∂β (cid:21) ∂ (cid:18) W ,β +1 ( t ) (cid:19) ∂β = (cid:16) t (cid:17) β (cid:20) ln (cid:16) t (cid:17) I β ( t ) + ∂I β ( t ) ∂β (cid:21) (13)7owever, di ff erentiation of the Bessel functions with respect to the order canbe expressed by [11] ∂J β ( t ) ∂β = π β (cid:82) π/ tan θ Y (cid:16) t [sin θ ] (cid:17) J β (cid:16) t [cos θ ] (cid:17) dθ ∂I β ( t ) ∂β = − β (cid:82) π/ tan θ K (cid:16) t [sin θ ] (cid:17) I β (cid:16) t [cos θ ] (cid:17) dθReβ > ff erentiation with respect to the parameter β can be ex-plicitly expressed [5] (cid:18) ∂J β ( t ) ∂β (cid:19) β =0 = π Y ( t ) (cid:18) ∂I β ( t ) ∂β (cid:19) β =0 = − K ( t ) (15)and therefore (cid:32) ∂ W ,β +1 ( − t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = − ln (cid:16) t (cid:17) J ( t ) + π Y ( t ) (cid:32) ∂ W ,β +1 ( t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = − ln (cid:16) t (cid:17) I ( t ) − K ( t ) (16)For β = 1 we have (cid:18) ∂J β ( t ) ∂β (cid:19) β =1 = J ( t ) t + π Y ( t ) (cid:18) ∂I β ( t ) ∂β (cid:19) β =1 = K ( t ) − I ( t ) t (17)which gives (cid:32) ∂ W ,β +1 ( − t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =1 = (cid:16) t (cid:17) (cid:104) − ln (cid:16) t (cid:17) J ( t ) − J ( t ) t + π Y ( t ) (cid:105)(cid:32) ∂ W ,β +1 ( t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =1 = (cid:16) t (cid:17) (cid:104) − ln (cid:16) t (cid:17) I ( t ) + K ( t ) − I ( t ) t (cid:105) (18)Derivatives for beta = 1 / (cid:18) ∂J β ( t ) ∂β (cid:19) β =1 / = (cid:113) πt [sin t Ci (2 t ) − cos t Si (2 t )] (cid:18) ∂I β ( t ) ∂β (cid:19) β =1 / = (cid:113) πt (cid:104) e t Ei ( − t ) − e − t Ei (2 t ) (cid:105) J / ( t ) = (cid:113) πt sin t ; I / ( t ) = (cid:113) πt sinh t (19)which leads to (cid:32) ∂ W ,β +1 ( − t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =1 / = √ πt (cid:104) − ln (cid:16) t (cid:17) sin t + sin t Ci (2 t ) − cos t Si (2 t ) (cid:105)(cid:32) ∂ W ,β +1 ( t ) ∂β (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =1 / = √ πt (cid:104) − ln (cid:16) t (cid:17) sinh t + (cid:16) e t Ei ( − t ) − e − t Ei (2 t ) (cid:17)(cid:105) (20)8f variable is changed to t = 2 x / , these results can be equivalently written indi ff erent form, for example (16) is (cid:18) ∂ W ,β +1 ( − x ) ∂β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = − ln √ x J (2 √ x ) + π Y (2 √ x ) (cid:18) ∂ W ,β +1 ( x ) ∂β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = − ln √ x I (2 √ x ) − K (2 √ x ) (21) ff erentiation of the Wright functionsof the second kind with respect to parameters We then consider among the Wright functions of the second kind the auxiliaryfunctions, F σ ( t ) and M σ ( t ) with σ ∈ (0 , ff usion processes in the 1990’s [4], defined as F σ ( t ) = W − σ, ( t ) ; 0 < σ < M σ ( t ) = W − σ, − σ ( t ) ; 0 < σ < F σ ( t ) = σ t M σ ( t ) (22)They series expansion explicitly read F σ ( t ) = ∞ (cid:88) k =1 ( − t ) k k ! Γ ( − σ k ) = 1 π ∞ (cid:88) k =1 ( − k +1 t k k ! Γ ( σ k + 1) sin( π σ k ) (23) M σ ( t ) = ∞ (cid:88) k =0 ( − t ) k k ! Γ ( − σ ( k + 1) + 1) = 1 π ∞ (cid:88) k =1 ( − t ) k − ( k − Γ ( σ k ) sin( π σ k ) (24)Direct di ff erentiation of (23) and (24) gives ∂F σ ( t ) ∂σ = π (cid:80) ∞ k =1 k ( − k +1 t k k ! Γ ( σ k + 1) [ ψ ( σ k + 1) sin( π σ k ) + π cos( π σ k )] ∂M σ ( t ) ∂σ = π (cid:80) ∞ k =1 k ( − t ) k − ( k − Γ ( σ k ) [ ψ ( σ k ) sin( π σ k ) + π cos( π σ k )] (25)Using the last equation in (22) we have ∂F σ ( t ) ∂σ = t M σ ( t ) + σ t ∂M σ ( t ) ∂σ (26)The second derivatives of these functions are ∂ F σ ( t ) ∂σ = π (cid:80) ∞ k =1 k ( − k +1 t k ( k − Γ ( σ k + 1) { [ ψ (cid:48) ( σ k + 1)+ ψ ( σ k + 1)] sin( π σ k )+2 π cos( π σ k ) ψ ( σ k + 1) − π sin( π σ k ) (cid:111) (27)and ∂ M σ ( t ) ∂σ = π (cid:80) ∞ k =1 k ( − t ) k − ( k − Γ ( σ k ) { [ ψ (cid:48) ( σ k )+ ψ ( σ k )] sin( π σ k )+2 π cos( π σ k ) ψ ( σ k ) − π sin( π σ k ) (cid:111) (28)9hey are interrelated by ∂ F σ ( t ) ∂σ = 2 t ∂M σ ( t ) ∂σ + σ t ∂ M σ ( t ) ∂σ (29) The Laplace transforms of the Wright functions are expressed in terms of two-parameter the Mittag-Le ffl er functions [3,4] L (cid:110) W α,β ( ± λt ) (cid:111) = 1 s E α,β (cid:18) ± λs (cid:19) ; α > λ > L (cid:110) e ± ρ W α,β ( λt ) (cid:111) = 1 s ∓ ρ E α,β (cid:32) λs ∓ ρ (cid:33) ; λ, ρ > L (cid:110) sinh( ρ t ) W α,β ( λt ) (cid:111) = (cid:110) s − ρ E α,β (cid:16) λs − ρ (cid:17) − s + ρ E α,β (cid:16) λs + ρ (cid:17)(cid:111) L (cid:110) cosh( ρ t ) W α,β ( λt ) (cid:111) = (cid:110) s − ρ E α,β (cid:16) λs − ρ (cid:17) + s + ρ E α,β (cid:16) λs + ρ (cid:17)(cid:111) (32)Multiplication (30) by t gives L (cid:110) t W α,β ( λt ) (cid:111) = − dds (cid:26) s E α,β (cid:18) λs (cid:19)(cid:27) = − (cid:40) − s E α,β (cid:18) λs (cid:19) + 1 s dds E α,β (cid:18) λs (cid:19)(cid:41) (33)Derivative of the Mittag-Le ffl er function is dds E α,β (cid:18) λs (cid:19) = − λs E α,β − (cid:16) λs (cid:17) − ( β − E α,β (cid:16) λs (cid:17) α (cid:16) λs (cid:17) (34)and finally we have L (cid:110) t W α,β ( λt ) (cid:111) = 1 s ( α λ − β + 1) E α,β (cid:16) λs (cid:17) + E α,β − (cid:16) λs (cid:17) α λ (35)In case that the Wright functions are expressed as the Bessel functions (see(2.18)), the Laplace transforms are known for β = 0 , , (cid:82) ∞ e − s t W , ( − λ t ) dt = (cid:82) ∞ e − s t J ( λt ) dt = √ s + λ (cid:82) ∞ e − s t W , ( − λ t ) dt = λ (cid:82) ∞ e − s t J ( λt ) t dt = s + √ s + λ ] (cid:82) ∞ e − s t W , ( − λ t ) dt = λ (cid:82) ∞ e − s t J ( λt ) t dt = λ (cid:40) λ [ s + √ s + λ ] + (cid:20) λ [ s + √ s + λ ] (cid:21) (cid:41) (36)10rom (12) it follows that (cid:90) ∞ e − s t W ,β +1 ( − λt ) dt = 1 λ β/ (cid:90) ∞ e − s t t − β/ J β (2 √ λt ) dt (37)and this integral equality is useful to derive explicit forms of the Mittag-Le ffl erfunctions. Starting with β = 0 we have [12] (cid:90) ∞ e − s t W , ( − λt ) dt = (cid:90) ∞ e − s t J (2 √ λt ) dt = 1 s e − λ/s ; Res > L (cid:8) W , ( − λt ) (cid:9) = 1 s E , ( − λs ) (39)and therefore by comparing the expected result is reached E , ( − λs ) = e − λ/s τ = λs E , ( − τ ) = e − τ (40)Introducing β = 1 into ( ?? ) we have [12] (cid:82) ∞ e − s t W , ( − λt ) dt = √ λ (cid:82) ∞ e − s t t − / J (2 √ λt ) dt = (cid:113) πλs e − λ/ s I / ( λ s ) = λ e − λ/ s sinh( λ s ) = λ (1 − e − λ/s ) (cid:82) ∞ e − s t W , ( − λt ) dt = s E , ( − λs ) τ = λs E , ( − τ ) = τ (1 − e − τ ) (41)In general case, the Laplace transform can be expressed in terms of the incom-plete gamma function γ ( a, z ) [12] (cid:82) ∞ e − s t W ,β +1 ( − λt ) dt = λ β/ (cid:82) ∞ e − s t t − β/ J β (2 √ λt ) dt = e i π β s β − λ β Γ ( β ) e − λ/s γ ( β, λs e − i π β ) ; Res > L (cid:110) W ,β +1 ( − λt ) (cid:111) = s E ,β +1 ( − λs ) = e i π β s β − λ β Γ ( β ) e − λ/s γ ( β, λs e − i π β ) z = λs E ,β +1 ( − z ) = e i π β Γ ( β ) z β e − z γ ( β, z e − i π β ) (43)For β = 2, we have exp( ± iπ ) and E , ( − z ) = 1 z e − z γ (2 , z ) = 1 z e − z (cid:90) z e − t t dt (44)11f n is positive integer, then γ ( n, z ) = Γ ( n ) P ( n, z ) P ( n, z ) = 1 − (cid:16) z + z + ... z n − ( n − (cid:17) e − z γ (2 , z ) = 1 − (1 + z ) e − z (45)There are some equivalent expressions in the form given in [12] E ,β +1 ( − z ) = e i π β Γ ( β ) z β e − z γ ( β, z e − i π β ) E ,β +1 ( − z ) = √ πe − z/ Γ ( β +1) z ( β +1) / M (1 − β ) / ,β/ ( z ) E ,β +1 ( − z ) = Γ ( β +1) 1 F (1; β + 1; − z ) = Γ ( β ) (cid:82) e z t (1 − t ) β − dt (46)For β being positive integer n , the last equation links the Mittag-Le ffl er func-tions with the Kummer functions (see also Appendix A in [14] for other re-sults).For positive values of argument t we have W ,β +1 ( t ) = t β / I β (2 √ t ) (47)and therefore (cid:90) ∞ e − s t W ,β +1 ( λt ) dt = 1 λ β/ (cid:90) ∞ e − s t t − β/ I β (2 √ λt ) dt (48)For β = 0, this gives [12] (cid:82) ∞ e − s t W , ( λt ) dt = (cid:82) ∞ e − s t I (2 √ λt ) dt = s e λ/s Res > L (cid:8) W , ( λt ) (cid:9) = 1 s E , ( λs ) (50)and by comparing the expected result is reached E , ( λs ) = e λ/s z = λs E , ( z ) = e z (51)If β = 1, then [12] (cid:82) ∞ e − s t W , ( λt ) dt = √ λ (cid:82) ∞ e − s t t − / I (2 √ λt ) dt = λ ( e λ/s − (cid:82) ∞ e − s t W , ( − λt ) dt = s E , ( λs ) z = λs E , ( z ) = z ( e z −
1) (52)12n general case [12] (cid:82) ∞ e − s t W ,β +1 ( λt ) dt = λ β/ (cid:82) ∞ e − s t t − β/ I β (2 √ λt ) dt = e λ/s s β − Γ ( β ) λ β γ ( β, λs ) L (cid:110) W ,β +1 ( λt ) (cid:111) = s E ,β +1 ( λs ) z = λs E ,β +1 ( z ) = e z z β γ ( β, z ) (53)where the incomplete gamma function can be expressed in terms of the Kum-mer function γ ( β, z ) = z β β e − z F (1; β + 1; z ) = z β β F (1; β + 1; − z ) (54)From (18) and (19) it follows that E ,β +1 ( z ) = e z z β γ ( β, z ) = 1 β F (1; β + 1; z ) = e z β F (1; β + 1; − z ) (55)Particular values of the incomplete gamma function of interest are γ (1 , z ) = (1 − e − z ) E , ( z ) = z ( e z −
1) (56)and γ (1 / , z ) = √ π erf ( √ z ) E , / ( z ) = (cid:112) πz e z erf ( √ z ) (57)In derivation explicit expressions for the Mittag-Le ffl er functions the recur-rence relation of the incomplete gamma function γ ( a + 1 , z ) = aγ ( a, z ) − z a e − z (58)is very useful. For example, for n = 1 , , γ (1 , z ) = z (1 − e − z ) γ (2 , z ) = γ (1 , z ) − z e − z = z (1 − e − z ) − z e − z γ (3 , z ) = γ (2 , z ) − z e − z = z (1 − e − z ) − z e − z γ ( n + 1 , z ) = n γ ( n, z ) − z e − z ; n = 1 , , , ... (59)and previously derived formulas in (51) and in (52) are reached.For n + 1 /
2, from (58) it follows that γ (1 / , z ) = √ π erf ( √ z ) γ (3 / , z ) = √ π erf ( √ z ) − z e − z γ (5 / , z ) = 2 [ √ π erf ( √ z ) − z e − z ] − z e − z (60)13nd immediately this gives by using (53) E , / ( z ) = (cid:112) πz e z erf ( √ z ) E , / ( z ) = e z z / [ √ π e z erf ( √ z ) − z e − z ] E , / ( z ) = e z z / (cid:110) √ π e z erf ( √ z ) − ze − z ] − z e − z (cid:111) (61)The Laplace transforms of the Mainardi auxiliary functions are L (cid:110) t F σ ( λt σ ) (cid:111) = L (cid:110) σ λt σ +1 M σ ( λt σ ) (cid:111) = e − λs σ < σ < λ > L (cid:110) σ F σ ( λt σ ) (cid:111) = L (cid:110) λt σ M σ ( λt σ (cid:111) = λs − σ e − λs σ < σ < λ > L (cid:110) σ W − σ, ( − λt σ ) (cid:111) = L (cid:110) λt σ W − σ, − σ ( − λt σ ) (cid:111) = λs − σ e − λs σ < σ < λ > σ = 1 / σ = 1 / σ = 1 / L (cid:110) t F / ( λt / ) (cid:111) = L (cid:110) λ t / M / ( λt / ) (cid:111) = e − λs / λ > L − (cid:110) e − λs / (cid:111) = λe − λ / t √ πt / t F / ( λt / ) = λ t / M / ( λt / ) = λe − λ / t √ πt / F / ( λτ ) = λτ M / ( λτ ) = λτ e − λ τ / √ π (65)Multiplication by t of the Mainardi function in (65) is equivalent to L (cid:110) F / ( λt / ) (cid:111) = L (cid:110) λ t / M / ( λt / ) (cid:111) = − dds (cid:110) e − λs / (cid:111) = λ s / e − λs / L − (cid:110) λ s / e − λs / (cid:111) = λe − λ / t √ π t F / ( λt / ) = λ t / M / ( λt / ) = λe − λ / t √ π t F / ( λτ ) = λτ M / ( λτ ) = λτ e − λ τ/ √ π (66)The same results, but in terms of the Wright functions can be written as L (cid:110) W − / , ( − λt / ) (cid:111) = L (cid:110) λt / W − / , / ( − λt / ) (cid:111) = λs / e − λs / L − (cid:110) λs / e − λs / (cid:111) = √ π t e − λ / t W − / , ( − λt / ) = λt / W − / , / ( − λt / ) = λ √ π t e − λ / t W − / , ( − λτ ) = λτ W − / , / ( − λτ ) = λτ √ π e − λ τ / (67)14n general case of the multiplication by t n , the di ff erentiation of exponentialfunctions can be expressed in terms of the Bessel functions [10] L (cid:110) t n F / ( λt / ) (cid:111) = L (cid:110) λ t n − / M / ( λt / ) (cid:111) = ( − n d n ds n (cid:110) e − λs / (cid:111) = λ n + 1 / s (1 − n ) / n − / √ π K n − / ( λ s / ) ( ?? ) (68)and therefore from (68) we have L (cid:110) t n W − / , ( − λt / ) (cid:111) = L (cid:110) λ t n − / W − / , / ( − λt / ) (cid:111) =( − n λ d n d s n (cid:26) e − λs / s / (cid:27) = λ n +3 / s − (2 n +1) / n − / √ π K n +1 / ( λs / ) (69)For σ = 1 /
3, we have L (cid:26) t F / ( λt / ) (cid:27) = L (cid:26) λ t / M / ( λt / ) (cid:27) = e − λs / (70)but using [15] L (cid:40) λ / π t / K / (cid:32) λ / √ t (cid:33)(cid:41) = e − λs / (71)we have 3 F / ( λt / ) = λt / M / ( λt / ) = λ / π t / K / ( 2 λ / √ t ) (72)The same result is available from [3,4] L (cid:26) F / ( λt / ) (cid:27) = L (cid:26) λt / M / ( λt / (cid:27) = λs / e − λs / (73)and [15] L (cid:40) λ / π t / K / (cid:32) λ / √ t (cid:33)(cid:41) = λe − λs / s / (74)In terms of the Wright functions it can be expressed by3 W − / , ( − λt / ) = λt / W − / , / ( − λt / ) = λ / π t / K / ( 2 λ / √ t ) (75) In 1969 Lamborn [9] proposed the following delta sequence for representationof the shifted Dirac delta function δ ( x −
1) = lim ν →∞ [ ν J ν ( νx )] (76)As it was demonstrated over the 2000-2008 period by Apelblat [16-18], thisdelta sequence is useful for evaluation of the asymptotic relations, limits of15eries, integrals and integral representations of elementary and special func-tions.If the Lamborn expression is multiplied by a function f ( tx ) and integratedfrom zero to infinity with respect to variable x we have f ( t ) = (cid:90) ∞ f ( t x ) δ ( x − dx = lim ν →∞ (cid:34) ν (cid:90) ∞ f ( t x ) J ν ( ν x ) dx (cid:35) (77)In such way, the function f ( t ) is represented by the asymptotic limit of theinfinite integral of product of f ( tx ) and the Bessel function J ν ( νx ). If the right-hand integral in (77) can be evaluated in the closed form, then the limit can beregarded as the generalization of the l’Hospital’s rule. f ( t ) = lim ν →∞ [ ν Φ ( t, ν )] Φ ( t, ν ) = (cid:82) ∞ f ( t x ) J ν ( ν x ) dx (78)For the Wright function treated as the generalized the Bessel function f ( t ) = W ,β +1 ( − t (cid:18) t (cid:19) β J β ( t ) (79)it follows from ((4.1) and (79) that f ( t ) = lim ν →∞ (cid:110) ν (cid:82) ∞ J ν ( ν x ) (cid:16) t x (cid:17) χ J β ( t x ) dx (cid:111) = W ,β +1 (cid:16) − t (cid:17) Φ ( t, ν, β ) = (cid:82) ∞ x − β J ν ( ν x ) J β ( t x ) dx (80)However, the infinite integral in (80) is known [19] Φ ( t, ν, µ ) = (cid:90) ∞ x − β J ν ( ν x ) J β ( t x ) dx = (cid:18) t (cid:19) β ν F (cid:32) ν + 12 , − ν β + 1; t ν (cid:33) (81)and therefore the Wright function is represented by the following limit W ,β +1 (cid:16) − t (cid:17) = lim ν →∞ (cid:110) F (cid:16) ν +12 , − ν ; β + 1 ; t ν (cid:17)(cid:111) Re ( ν + 1) > Reβ > − < t < ν (82)or in the equivalent form W ,β +1 ( − x ) = lim ν →∞ (cid:26) F (cid:18) ν + 12 , − ν β + 1 ; 4 xν (cid:19)(cid:27) (83)For β = 0, we have W , (cid:32) − t (cid:33) = J ( t ) = lim ν →∞ (cid:40) F (cid:32) ν + 12 , − ν t ν (cid:33)(cid:41) (84)and for β = ± / W , / (cid:16) − t (cid:17) = (cid:113) t J / ( t ) = lim ν →∞ (cid:110) F (cid:16) ν +12 , − ν ; ; t ν (cid:17)(cid:111) = t √ π t W , / (cid:16) − t (cid:17) = (cid:113) t J − / ( t ) = lim ν →∞ (cid:110) F (cid:16) ν +12 , − ν ; ; t ν (cid:17)(cid:111) = − cos t √ π (85)16ypergeometric functions (85) are known in di ff erent form [20] F (cid:16) a, − a ; ; (sin z ) (cid:17) = sin[(2 a − z ](2 a − z F (cid:16) a, − a ; ; (sin z ) (cid:17) = cos[(2 a − z ]cos z a = ν +12 ; sin z = tν (86)If the delta sequence in (76) is used together with integral transforms hav-ing di ff erent kernels T , the we have [17,18]lim ν →∞ [ ν (cid:82) ∞ f ( ξ, λ ) T { J ν ( ν x ) , ξ } dξ ] =lim ν →∞ [ ν (cid:82) ∞ J ν ( ν x ) T { f ( ξ, λ ) , x } dx ] = T (1 , λ ) (87)In the case of the Laplace transformation, (87) can be written in the followingway (cid:82) ∞ e − ξ x J ν ( ν ξ ) dξ = ν ν √ ν + ξ [ ξ + √ ν + ξ ] ν lim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ f ( ξ,λ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = L (1 , λ ) (88)Introducing (30) into (88) we havelim ν →∞ ν ν +1 (cid:90) ∞ W α,β ( λξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = 1 s E α,β ( λs ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =1 = E α,β ( λ ) (89)The same operation performed with (31) giveslim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ e ρ ξ W α,β ( λξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = − ρ E α,β ( λ − ρ )lim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ e − ρ ξ W α,β ( λξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = ρ E α,β ( λ ρ ) (90)and using (35) lim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ ξ W α,β ( λξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = α λ [( α λ − β + 1) E α,β ( λ ) + E α,β − ( λ )] (91)The Laplace transform in (53) leads tolim ν →∞ ν ν +1 (cid:90) ∞ W ,β +1 ( λξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = e − λ λ β Γ ( β ) γ ( β, λ ) = E ,β +1 ( λ )(92)For integer values of parameters β in (92), the limits of the Wright functionscan be represented by simple expressions. For β = 1, we havelim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ W , ( λξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = e λ lim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ W , ( − λξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = e − λ (93)17nd lim ν →∞ ν ν +1 (cid:90) ∞ W , ( − λ ξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = 1 √ λ (94)For β = 1, the corresponding limits arelim ν →∞ ν ν +1 (cid:90) ∞ W , ( − λξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = 2 λ e − λ sinh( λ ν →∞ ν ν +1 (cid:90) ∞ W , ( λξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = 1 λ ( e λ −
1) (96)lim ν →∞ ν ν +1 (cid:90) ∞ W , ( − λ ξ ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = 21 + √ λ (97)Similarly, for β = 3, the functional limit islim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ W , ( − λ ξ ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = λ (cid:40) λ √ λ + (cid:20) λ √ λ (cid:21) (cid:41) (98)The Laplace transforms of the Mainardi functions from (68) and (69) arelim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ ξ n F / ( λξ / ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = λ n +1 n − / √ π K n − / ( λ )lim ν →∞ (cid:40) ν ν +1 (cid:82) ∞ ξ n − / M / ( λξ / ) √ ν + ξ [ ξ + √ ν + ξ ] ν dξ (cid:41) = λ n − / n +1 / √ π K n − / ( λ ) (99)and from (73) we havelim ν →∞ ν ν +1 (cid:90) ∞ F / ( λξ / ) (cid:112) ν + ξ [ ξ + (cid:112) ν + ξ ] ν dξ = λ e − λ (100) Parameters of the Wright functions were treated as variables and derivativeswith respect to them were derived and discussed. These derivatives are ex-pressible in terms of infinite power series with quotients of digamma andgamma functions in their coe ffi cients. The functional form of these series re-sembles those which were derived for the Mittag-Le ffl er functions. Only in fewcases, it was possible to obtain the sums of these series in a closed form. Thedi ff erentiation operation when the Wright functions are treated as the gener-alized Bessel functions leads to the Bessel functions and their derivatives withrespect to the order. Simple operations with the Laplace transforms of the18right functions of the first kind give explicit forms of the Mittag-Le ffl er func-tions. Applying the shifted Dirac delta function, permits to derive functionallimits by using the Laplace transforms of the Wright functionsFinally we would like to draw attention of the interested readers to therecent contributions where Wright functions are discussed: in particular [21],[22] for some applications in fractional di ff erential equations of the Wrightfunctions of the first and second kind, respectively. and [23] for asymptotics ofWright functions of the second kind. Acknowledgements
We are grateful to Associate Professor Juan Luis Gonzalez-Santander Martinez,Department of Mathematics, Universidad de Oviedo, Oviedo, Spain for hishelp with using MATHEMATICA program and with editing in LaTeX.
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