MSC 2010: 30A10, 30B10, 30B30, 30B50, 30D15, 33E12In studying the behaviour of series, defined by means of the Mittag-Leffler functions, on the boundary of its domain of convergence in the complex plane, we prove Cauchy-Hadamard, Abel, Tauber and Littlewood type theorems. Asymptotic formulae are also provided for the Mittag-Leffler functions in the case of \large" values of indices that are used in the proofs of the convergence theorems for the considered series

Paneva-Konovska, Jordanka

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

SERIES IN MITTAG-LEFFLER FUNCTIONS:INEQUALITIES AND CONVERGENT THEOREMSJordanka Paneva-KonovskaThis paper is dedicated to the 70th anniversary of Professor SrivastavaAbstractIn studying the behaviour of series, defined by means of the Mittag-Leﬄer functions, on the boundary of its domain of convergence in the com-plex plane, we prove Cauchy-Hadamard, Abel, Tauber and Littlewood typetheorems. Asymptotic formulae are also provided for the Mittag-Leﬄerfunctions in the case of “large” values of indices that are used in the proofsof the convergence theorems for the considered series.MSC 2010 : 30A10, 30B10, 30B30, 30B50, 30D15, 33E12Key Words and Phrases: Mittag-Leﬄer functions, inequalities, asymp-totic formula, Cauchy-Hadamard, Abel, Tauber and Littlewood type theo-rems, summation of divergent series1. IntroductionThe Mittag-Leﬄer functions Eα (Mittag-Leﬄer, 1902-1905) and Eα, β(Agarwal 1953, see also [5]), are defined in the whole complex plane C bythe power series:Eα(z) =∞∑k=0zkΓ(αk + 1), Eα,β(z) =∞∑k=0zkΓ(αk + β), α > 0, β > 0. (1.1)c© 2010, FCAA – Diogenes Co. (Bulgaria). All rights reserved.404 J. Paneva-KonovskaThe Mittag-Leﬄer functions (1.1) are examples of entire functions ofa given order ρ = 1/α and a type σ = 1. They have been studied indetails by Dzrbashjan [1], [2]: asymptotic formulae in different parts of thecomplex plane, distribution of the zeros, kernel functions of inverse Boreltype integral transforms, various relations and representations. The detailedproperties of these functions can be found in the contemporary monographsof Kilbas et al. [4] and Podlubny [15].In our previous papers ([9] - [12]) we studied series in systems of someother representatives of the SF of FC, which are fractional indices analoguesof the Bessel functions and also multi-index Mittag-Leﬄer functions (in thesense of [5],[6],[7]), and proved Cauchy-Hadamard, Abel and Tauberian typetheorems in the complex domain.In this paper we prove some inequalities in the complex plane C and onits compact subsets, asymptotic formulae for ”large” values of indices of thefunctions (1.1) and study the convergence of series in such kind of functions.2. Inequalities and asymptotic formulaeIn this point we prove some asymptotic formulae for ”large” values ofindices. Denoteθn(z) =∞∑k=1zkΓ(kn+ 1), θn,β(z) = Γ(β)∞∑k=1zkΓ(kn+ β), (2.1)θα,n(z) = Γ(n)∞∑k=1zkΓ(αk + n). (2.2)Lemma 2.1. Let n ∈ N, z ∈ C and K ⊂ C be a nonempty compact set.Then the following inequalities hold|θn(z)| ≤ 1n!( exp (|z|)− 1) , |θn,β(z)| ≤ 1(n− 1)!|z| exp(|z|)β, (2.3)|θα,n(z)| ≤ Γ(n)Γ(α+ n) Γ(α+ 1) (Eα(|z|)− 1) , (2.4)and moreover there exists a constant C, 0 < C <∞, such that|θn(z)| ≤ C/n!, |θn,β(z)| ≤ C/(n− 1)!, |θα,n(z)| ≤ C Γ(n)Γ(α+ n) , (2.5)for all the natural numbers n and each z ∈ K.SERIES IN MITTAG-LEFFLER FUNCTIONS: . . . 405P r o o f. First, let z ∈ C. Then we can writeθn(z) =1Γ(n+ 1)∞∑k=1Γ(n+ 1)Γ(kn+ 1)zk =1n!∞∑k=1n!(kn)!zk.Denoting uk(z) =n!(kn)!zk, we obtain the estimate |uk(z)| ≤ |z|kk!for theabsolute value of uk(z). Since the series∞∑k=1|z|kk!converges for each z ∈ Cand its sum is exp(|z|)− 1, then the first of the estimates (2.3) hold on thewhole complex plane.The second of the estimates (2.3) are proved in [13].To prove (2.4), we writeθα,n(z) =Γ(n)Γ(α+ n)∞∑k=1Γ(α+ n)Γ(αk + n)zkand denotingγ˜n,k =Γ(α+ n)Γ(αk + n), u˜n,k(z) = γ˜n,k zk,we obtain consecutivelyγ˜n,1 = 1; 0 < γ˜n,k =Γ(α+ 1)Γ(αk + 1)n−1∏s=1(α+ s)(αk + s)≤ Γ(α+ 1)Γ(αk + 1), for k ∈ N, k 6= 1,|u˜n,k(z)| = γ˜n,k |z|k ≤ Γ(α+ 1)Γ(αk + 1) |z|k, for k ∈ N,and therefore|θα,n(z)| ≤ Γ(n)Γ(α+ 1)Γ(α+ n)( ∞∑k=0|z|kΓ(αk + 1)− 1)which proves (2.4).Further, for all z on the compact set K, the inequalities (2.5) followimmediately from the inequalities (2.3) and (2.4).Theorem 2.1. For the Mittag-Leﬄer functions En, En,β, Eα,n (n ∈ N),the following asymptotic formulaeEn(z) = 1 + θn(z), z ∈ C, θn(z)→ 0 as n→∞ (2.6)406 J. Paneva-KonovskaEn,β(z) =1Γ(β)(1 + θn,β(z)), z ∈ C, θn,β(z)→ 0 as n→∞ (2.7)Eα,n(z) =1Γ(n)(1 + θα,n(z)), z ∈ C, θα,n(z)→ 0 as n→∞ (2.8)are valid. The functions θn(z), θn,β(z), θα,n(z) are holomorphic for z ∈ C.The convergence is uniform on the compact subsets of C.P r o o f. The identities (2.6)-(2.8) obtain due to (1.1), (2.1) and (2.2)automatically. The holomorphy of θn(z), θn,β(z), θα,n(z) follows from theholomorphy of En(z), En,β(z), Eα,n(z) on the whole complex plane and theequalities (2.6) - (2.8). The rest follows immediately from Lemma 2.1.Note 2.1. According to the asymptotic formulae (2.6) - (2.8), it followsthere exists a natural number M such that the functions En, Γ(n)Eα,n,Γ(β)En,β have not any zeros at all for n > M .3. Series in Mittag-Leﬄer functionsWe introduce the following auxiliary functions, related to Mittag-Leﬄer’sfunctions, adding E˜0(z), E˜0,β(z)) and E˜α,0(z) just for completeness, namely:E˜0(z) = 1; E˜n(z) = znEn(z), n ∈ N,E˜0,β(z) = 1; E˜n,β(z) = Γ(β)znEn,β(z), n ∈ N; β > 0,E˜α,0(z) = 1; E˜α,n(z) = Γ(n)znEα,n(z), n ∈ N; α > 0,and consider the series in these functions, respectively:∞∑n=0anE˜n(z),∞∑n=0anE˜n,β(z),∞∑n=0anE˜α,n(z), (3.1)with complex coefficients an (n = 0, 1, 2, ...).Our main objective is to study the convergence of the series (3.1) in thecomplex plane. We prove theorems, corresponding to the classical Cauchy-Hadamard, Abel, Tauber and Littlewood theorems. Such kind of results areprovoked by the fact that the solutions of some fractional order differentialand integral equations can be written in terms of series (or series of integrals)of Mittag-Leﬄer functions (as for example in Saqabi and Kiryakova [19]).Convergence theorems are obtained also for series in other special functions,for example, for series in Laguerre and Hermite polynomials [16] - [18], andresp. by the author for series in Bessel functions and their Wright’s 2, 3,and 4-index generalizations in the previous papers [9] - [12].SERIES IN MITTAG-LEFFLER FUNCTIONS: . . . 4074. Cauchy-Hadamard and Abel type theoremsIn the beginning we give a theorem of Cauchy-Hadamard type for everyone of the above series.Theorem 4.1. (of Cauchy-Hadamard type). The domain of con-vergence of each one of the series (3.1) with complex coefficients an is thedisk |z| < R with a radius of convergence R = 1/Λ, whereΛ = lim supn→∞( |an| )1/n. (4.1)The cases Λ = 0 and Λ = ∞ can be included in the general case, provided1/Λ means ∞, respectively 0.I d e a o f P r o o f. Using the asymptotic formulae (2.6) - (2.8), weevaluate the absolute value of the general term of each of the series (3.1).Further the proof goes separately in the three cases: Λ = 0, 0 < Λ <∞, Λ = ∞. We show the absolute convergence of the series (3.1) in thecircular domain {z : z ∈ C, |z| < R}. In the second case we prove thatthe series is divergent for |z| > R and in the third case - divergent for allcomplex z 6= 0.Let z0 ∈ C, 0 < R < ∞, |z0| = R and gϕ be an arbitrary angulardomain with size 2ϕ < pi and with vertex at the point z = z0, which issymmetric with respect to the straight line defined by the points 0 and z0.The following theorem is valid.Theorem 4.2. (of Abel type). Let {an}∞n=0 be a sequence of complexnumbers, Λ be the real number defined by (4.1), 0 < Λ <∞. Let K = {z :z ∈ C, |z| < R,R = 1/Λ}. If f(z), g(z;β), h(z;α) are the sums respectivelyof the first, second and third of the series (3.1) on the domain K, and theseseries converge at the point z0 of the boundary of K, then:limz→z0f(z) =∞∑n=0anE˜n(z0), limz→z0g(z; β) =∞∑n=0anE˜n, β(z0), (4.2)limz→z0h(z; α) =∞∑n=0anE˜α, n(z0), (4.3)provided |z| < R and z ∈ gϕ.408 J. Paneva-KonovskaP r o o f. Let us consider the difference∆(z) =∞∑n=0anE˜n(z0)− f(z) =∞∑n=0an(E˜n(z0)− E˜n(z)) (4.4)and represent it in the form∆(z)=k∑n=0an(E˜n(z0)− E˜n(z)) +∞∑n=k+1an(E˜n(z0)− E˜n(z)).Let p > 0. By using the notationsβm =m∑n=k+1anE˜n(z0), m > k, βk = 0, γn(z) = 1− E˜n(z)/E˜n(z0),and the Abel transformation (see in ([8], vol.1, ch.1, p.32, 3.4:7), we obtainconsecutively:k+p∑n=k+1an(E˜n(z0)− E˜n(z)) =k+p∑n=k+1(βn − βn−1)γn(z)= βk+pγk+p(z)−k+p−1∑n=k+1βn(γn+1(z)− γn(z)),i.e.k+p∑n=k+1an(E˜n(z0)− E˜n(z)) = (1− E˜k+p(z)/E˜k+p(z0))k+p∑n=k+1anE˜n(z0)−k+p−1∑n=k+1(n∑s=k+1asE˜s(z0))(E˜n(z))E˜n(z0))− E˜n+1(z))E˜n+1(z0))).According to Note 2.1, there exists a natural numberM such that E˜n(z0) 6=0when n > M . Let k > M . Then, for every natural n > k:E˜n(z)/E˜n(z0)− E˜n+1(z)/E˜n+1(z0) (4.5)= (z/z0)n(1+θn(z))(1+θn+1(z0))−(z/z0)(1+θn+1(z))(1+θn(z0))(1+θn(z0))(1+θn+1(z0))·For the right hand side of (4.5) we apply the Schwartz lemma. Then we getthat there exists a constant C:|E˜n(z)/E˜n(z0)− E˜n+1(z)/E˜n+1(z0)| ≤ C|z − z0||z/z0|n.Analogously, there exists a constant B:|1− E˜k+p(z)/E˜k+p(z0)| ≤ B|z − z0| ≤ 2B|z0|.Let ε be an arbitrary positive number and choose N(ε) so large that fork > N(ε) the inequalitySERIES IN MITTAG-LEFFLER FUNCTIONS: . . . 409|n∑s=k+1asE˜s(z0)| < min(ε cosϕ/(12B|z0|), ε cosϕ/(6C|z0|))holds for every natural n > k. Therefore, for k > max(M,N(ε)):|∞∑s=k+1asE˜s(z0)| ≤ min(ε cosϕ/(12B|z0|), ε cosϕ/(6C|z0|),and|∞∑n=k+1an(E˜n(z0)− E˜n(z))| ≤ (ε cosϕ/6)(1 +∞∑n=k+1|z0|−1|z − z0||z/z0|n)≤ (ε cosϕ/6)(1 + |z − z0|/(|z0| − |z|)).But near the vertex of the angular domain gϕ in the part dϕ closed betweenthe angle’s arms and the arc of the circle with center at the point 0 andtouching the arms of the angle, we have |z − z0|/(|z0| − |z|) < 2/ cosϕ, i.e.|z − z0| cosϕ < 2(|z0| − |z|). That is why the inequality|∞∑n=k+1an(E˜n(z0)− E˜n(z))| < (ε cosϕ)/6 + ε/3 ≤ ε/2 (4.6)holds for z ∈ dϕ and k > max(M,N(ε)). Fix some k > max(M,N(ε)) andafter that choose δ(ε) such that if |z − z0| < δ(ε) then the inequality|k∑n=0an(E˜n(z0)− E˜n(z))| < ε/2 (4.7)holds inside dϕ. We get|∆(z)| = |∞∑n=0an(E˜n(z0)− E˜n(z))|for the module of the difference (4.4). From (4.6) and (4.7) it follows thatthe first of the equalities (4.2) is satisfied.The proofs of the second of the equalities (4.2) and (4.3) go by analogy.5. (E, z0) - summationsLet us consider the numerical series∞∑n=0an, an ∈ C, n = 0, 1, 2, ... (5.1)410 J. Paneva-KonovskaTo define its Abel summability ([3], p.20, 1.3 (2)), we consider also thepower series∞∑n=0anzn.Definition 5.1. The series (5.1) is called A - summable, if the series∞∑n=0anzn converges in the disk D = {z : z ∈ C, |z| < 1} and moreover thereexists limz→1−0∞∑n=0anzn = S.The complex number S is called A-sum of the series (5.1) and the usualnotation of that is∞∑n=0an = S (A).Note 5.1. The A-summation is regular. It means that if the series(5.1) converges, then it is A-summable, and its A-sum is equal to its usualsum.Note 5.2. The A-summability of the series (5.1) does not imply ingeneral its convergence. But, with additional conditions on the growth ofthe general term of the series (5.1), the convergence can be ensured.Note that each of the functions E˜n(z), E˜n,β(z), E˜α,n(z), (n ∈ N), beingan entire function, no identically zero, has no more than finite number ofzeros in the closed and bounded set |z| ≤ R ([8], Vol. 1, Ch. 3, §6, 6.1,p.305). Moreover, because of Note 2.1, no more than finite number of thesefunctions have some zeros.Let z0 ∈ C, |z0| = R, 0 < R < ∞, E˜n(z0) 6= 0, E˜n,β(z0) 6= 0, andE˜α,n(z0) 6= 0. For the sake of brevity, denoteE∗n(z; z0) =E˜n(z)E˜n(z0), E∗n,β(z; z0) =E˜n,β(z)E˜n,β(z0), E∗α,n(z; z0) =E˜α,n(z)E˜α,n(z0).Definition 5.2. The series (5.1) is said to be (E, z0) - summable, ifthe series ∞∑n=0anE∗n(z; z0), (5.2)converges in the disk |z| < R and, moreover, there exists the limitlimz→z0∞∑n=0anE∗n(z; z0), (5.3)SERIES IN MITTAG-LEFFLER FUNCTIONS: . . . 411provided z remains on the segment [0, z0).Definition 5.3. The series (5.1) is said to be (Eβ, z0) - summable(respectively (Eα, z0) - summable), if the series∞∑n=0anE∗n,β(z; z0) (respectively∞∑n=0anE∗α,n(z; z0))converges in the disk |z| < R and, moreover, there exists the limitlimz→z0∞∑n=0anE∗n,β(z; z0) (respectively limz→z0∞∑n=0anE∗α,n(z; z0)),provided z remains on the segment [0, z0).Note 5.3. Every (E, z0)−, (Eβ, z0)−, (Eα, z0)− summation is regular,and this property is just a particular case of Theorem 4.2.6. Tauberian type theoremsA Tauberian theorem is a statement that relates the Abel summabil-ity and the standard convergency of a numerical series by means of someassumptions imposed on the general term of the series under question. Aclassical result in this direction is given by Theorem 85 ([3]).In this paper we extend the validity of such type of assertion to seriesin Mittag-Leﬄer functions. Tauber type theorems are given also for sum-mations by means of Laguerre polynomials [16], and Bessel type functionsby the author [9] - [12].Theorem 6.1. (of Tauber type). If the series (5.1) is (E, z0) -summable, (or (Eβ, z0) respectively (Eα, z0) - summable), andlimn→∞nan = 0, (6.1)then it is convergent.P r o o f. Let z belong to the segment [0, z0]. Taking into account theasymptotic formula (2.6) for the Mittag-Leﬄer functions, we obtain:anE∗n(z; z0) = an(zz0)n 1 + θn(z)1 + θn(z0)= an(zz0)n (1 + θ˜n(z; z0)),where θ˜n(z; z0) =θn(z)− θn(z0)1 + θn(z0). Then, due to (2.5):θ˜n(z; z0) = O(1/n!). (6.2)412 J. Paneva-KonovskaLet us write (5.2) in the form∞∑n=0anE∗n(z; z0) =∞∑n=0an(zz0)n (1 + θ˜n(z; z0)). (6.3)Denoting wn(z) = an(zz0)nθ˜n(z; z0) we consider the series∞∑n=0wn(z). Since|wn(z)| ≤ |an| |θ˜n(z; z0)| and according to condition (6.1) and the relation-ship (6.2), there exists a constant C, such that |wn(z)| ≤ C/n2. Since∞∑n=11/n2 converges, the series∞∑n=0wn(z) is also convergent, even absolutelyand uniformly on the segment [0, z0]. Therefore (since limz→z0wn(z) = 0)limz→z0∞∑n=0wn(z) =∞∑n=0limz→z0wn(z) = 0.Obviously, the assumption that the series (5.1) is (E, z0) - summable impliesthe existence of the limit (5.3). Then, having in mind that (6.3) can bewritten in the form∞∑n=0anE∗n(z; z0) =∞∑n=0an(zz0)n+∞∑n=0an(zz0)nθ˜n(z; z0),we conclude that there exists the limitlimz→z0∞∑n=0an(zz0)n(6.4)and, moreover,limz→z0∞∑n=0anE∗n(z; z0) = limz→z0∞∑n=0an(zz0)n.From the existence of the limit (6.4) it follows that the series (5.1) is A-summable. Then according to Theorem 85 from [3], the series (5.1) con-verges.The proofs of the theorem for the other two cases (Eβ, z0)- and (Eα, z0)-summations go in the same way.At first sight it seems that the condition an = o(1/n) is essential. Nev-ertheless, Littlewood succeeds to weaken it and obtain the strengthenedversion of the Tauber theorem ([3], Theorem 90).A Littlewood generalization of the o(n) version of a Tauber type theorem(Theorem 6.1) is also given in this part. Similar theorem for series in Besselfunctions series is also proved, see [12].Theorem 6.2 (of Littlewood type). If the series (5.1) is (E, z0) -summable, (or (Eβ, z0) respectively (Eα, z0) - summable), andSERIES IN MITTAG-LEFFLER FUNCTIONS: . . . 413an = O(1/n) (6.5)then the series (5.1) converges.I d e a o f t h e P r o o f. Using Theorem 90 in the place ofTheorem 85, [3], the proof of the (Eβ, z0) summability follows the line ofthat of Theorem 6.1 and the ideas of the proofs of the cases of (E, z0)- and(Eα, z0)-summabilities, given in [13], respectively [14].AcknowledgementsThis paper is partially supported by Project D ID 02/25/2009: “IntegralTransform Methods, Special Functions and Applications”, National ScienceFund - Ministry of Education, Youth and Science, Bulgaria.This work is performed in the frames of the Research Project “TransformMethods, Special Functions, Operational Calculi and Applications” underthe auspices of the Institute of Mathematics and Informatics - BulgarianAcademy of Sciences.References[1] M.M. Dzrbashjan, Integral Transforms and Representations in theComplex Domain (in Russian). Nauka, Moskow, 1966.[2] M.M. Dzrbashjan, Interpolation and Spectrum Expansions Associatedwith Fractional Differential Operators (in Russian). Erevan, 1983.[3] G. Hardy, Divergent Series. Oxford, 1949.[4] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applicationsof Fractional Differential Equations, Elsevier, 2006.[5] V. Kiryakova, Multiple (multiindex) Mittag-Leﬄer functions and rela-tions to generalized fractional calculus. Journal of Comput. and Appl.Mathematics 118 (2000), 241-259.[6] V. Kiryakova, The multi-index Mittag-Leﬄer functions as im-portant class of special functions of fractional calculus. Comput-ers and Mathematics with Appl., 59, No 5 (2010), 1885-1895,doi:10.1016/j.camwa.2009.08.025[7] V. Kiryakova, The special functions of fractional calculus as gener-alized fractional calculus operators of some basic functions, Com-puters and Mathematics with Appl. 59, No 3 (2010), 1128-1141,doi:10.1016/j.camwa.2009.05.014.[8] A. Markushevich, A Theory of Analytic Functions, Vols. 1, 2 (In Rus-sian). Nauka, Moscow, 1967.414 J. Paneva-Konovska[9] J. Paneva-Konovska, On the convergence of series in Bessel-Maitlandfunctions. Annuaire de l’ Univ. de Sofia, Faculte de Math. et Informa-tique 99 (2009), 75-84.[10] J. Paneva-Konovska, Cauchy-Hadamard, Abel and Tauber type theo-rems for series in generalized Bessel-Maitland functions. Compt. Rend.Acad. Bulg. Sci. 61, No 1 (2008), 9-14.[11] J. Paneva-Konovska, Theorems on the convergence of series in gen-eralized Lommel-Wright functions. Fract. Calc. Appl. Anal. 10, No 1(2007), 59-74.[12] J. Paneva-Konovska, Tauberian theorem of Littlewood type for seriesin Bessel functions of first kind. Compt. Rend. Acad. Bulg. Sci. 62, No2 (2009), 75-84.[13] J. Paneva-Konovska, Convergence of series in Mittag-Leﬄer functions.Compt. Rend. Acad. bulg. Sci. 63, No 6 (2010), 815-822.[14] J. Paneva-Konovska, Convergence of series in some multi-indexMittag-Leﬄer functions. Proc. of FDA’10, 4th IFAC Workshop Frac-tional Differentiation and its Applications, Badajoz, Spain, Oct. 18-20,2010 (Eds: I. Podlubny, B. M. Vinagre Jara, YQ. Chen, V. Feliu Batlle,I. Tejado Balsera), Article No FDA10-147, 1-4.[15] I. Podlubny, Fractional Differential Equations. Acad. Press, 1999.[16] P. Rusev, A theorem of Tauber type for the summation by meansof Laguerre’s polynomials. Compt. Rend. Acad. Bulg. Sci. 30, No 3(1977), 331-334 (in Russian).[17] P. Rusev, Analytic Functions and Classical Orthogonal Polynomials.Publ. House Bulg. Acad. Sci., Sofia, 1984.[18] P. Rusev, Classical Orthogonal Polynomials and Their AssociatedFunctions in Complex Domain. Publ. House Bulg. Acad. Sci., Sofia,2005.[19] B. Al-Saqabi and V. Kiryakova, Transmutation method for solvingErdelyi-Kober fractional differintegral equations. J. Math. Anal. Appl.211 (1997), 347-364.Faculty of Applied Mathematics and InformaticsTechnical University of Sofia8 Kliment Ohridski, 1000 Sofia, BULGARIAe-mail: yorry77@mail.bg Received: GFTA, August 27-31, 2010

Series in Mittag-Leffler Functions: Inequalities and Convergent Theorems

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Series in Mittag-Leffler Functions: Inequalities and Convergent Theorems

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