Mathematics Subject Classification: 33C05, 33C10, 33C20, 33C60, 33E12,
33E20, 40A30The main purpose of this paper is to present a number of potentially
useful integral representations for the generalized Mathieu series as well as
for its alternating versions via Mittag-Leffler type functions

Tomovski, Živorad

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

INTEGRAL REPRESENTATIONS OFGENERALIZED MATHIEU SERIESVIA MITTAG-LEFFLER TYPE FUNCTIONSZˇivorad TomovskiAbstractThe main purpose of this paper is to present a number of potentiallyuseful integral representations for the generalized Mathieu series as well asfor its alternating versions via Mittag-Leﬄer type functions.Mathematics Subject Classification: 33C05, 33C10, 33C20, 33C60, 33E12,33E20, 40A30Key Words and Phrases: integral representations, Mathieu series, Mittag-Leﬄer functions, Laplace transform1. IntroductionThe following infinite series:S(r) =∞∑n=12n(n2 + r2)2(r ∈ R+) (1.1)is known as named after Emile Leonard Mathieu (1835-1890), who investi-gated it in his 1890 work [11] on elasticity of solid bodies.A remarkable useful integral representation for S(r) in the elegant form:S(r) =1r∞∫0x sin (xr)ex − 1 dx (1.2)128 Zˇ. Tomovskiwas given by Emersleben [6].Several interesting problems and solutions dealing with integral repre-sentations and bounds for the following slight generalization of the Mathieuseries with a fractional power :Sµ(r) =∞∑n=12n(n2 + r2)µ(r ∈ R+; µ > 1) (1.3)can be found in the recent works by Diananda [4], Tomovski and Trencˇevski[21] and Cerone and Lenard [3]. Motivated essentially by the works ofCerone and Lenard [3] (and Qi [17]), we defined in [18] a family of generalizedMathieu series:S(α,β)µ (r;a) = S(α,β)µ (r; {ak}∞k=1) =∞∑n=12aβn(aαn + r2)µ(r, α, β, µ ∈ R+), (1.4)where limk→∞ak =∞.Comparing the definitions (1.1), (1.3) and (1.4), we see that S2(r) = S(r)and Sµ(r) = S(2,1)µ (r, {k}∞k=1).Furthermore, the special casesS(2,1)2 (r, {ak}∞k=1), S(2,1)µ (r, {kγ}∞k=1) and S(α,α/2)µ (r, {k}∞k=1)were investigated by Qi [17], Tomovski [20] and Cerone and Lenard [3].In [18] the following integral representations were proposed:S(α,β)µ (r; {kγ}∞k=1)=2Γ(µ)∞∫0xγ(µα−β)−1ex − 1 1Ψ1[(µ, 1); (γ(µα− β), γα) ;−r2xγα] dx (1.5)(r, α, β, γ ∈ R+; γ(µα− β) > 1);S(α,β)µ (r; {kq/α})=2Γ(q[µ− βα ])∞∫0xq[µ−βα]−1ex − 1 1Fq(µ;4(q; q[µ− βα]);−r2(xq)q)dx(1.6)(r, α, β ∈ R+;µ− βα> q−1; q ∈ N),where 4(q;λ) is the q−tuple λq,λ+ 1q, . . . ,λ+ q − 1q;INTEGRAL REPRESENTATIONS OF . . . 129S(α,α/2)µ+1 (r; {k2/α}∞k=1) = S(2,1)µ+1 (r; {k}∞k=1) = Sµ+1(r)=√pi(2r)µ−12Γ(µ+ 1)∞∫0xµ+12ex − 1Jµ− 12 (rx)dx, (r, µ ∈ R+); (1.7)S(α,0)µ (r; {k2/α}∞k=1) =∞∑n=12(n2 + r2)µ=2√pi(2r)µ−12Γ(µ)∞∫0xµ−12ex − 1Jµ− 12 (rx)dx, (r ∈ R+; µ >12). (1.8)Here pΨq denotes the Fox-Wright generalization of the hypergeometric pFqfunction with p numerator and q denominator parameters (see for example[7], v.1; [19]), p.50, eq. 1.5 (21); [10], Appendix) and Jν – the Bessel functionof first kind.In [14] we introduced an alternating variant of S(r),S˜(r) =∞∑n=1(−1)n−1 2n(n2 + r2)2(r ∈ R+), (1.9)called alternative Mathieu series.Using the formulas given in [12]:∞∑k=1L′(k) = −∞∫0tf(t)et − 1dt, (1.10)∞∑k=1(−1)k−1L′(k) =∞∫0tf(t)et + 1dt, (1.11)where L(k) is the classical Laplace transform of f(t), we obtained in [14]the integral representation of S˜(r):S˜(r) =1r∞∫0x sin (xr)ex + 1dx. (1.12)Let S˜(α,β)µ (r;a) be alternating variant of (1.4). Integral representationsfor the alternating variant of (1.8) are given in [14] by:130 Zˇ. TomovskiS˜(α,β)µ (r; {kγ}∞k=1)=2Γ(µ)∞∫0xγ(µα−β)−1ex + 1 1Ψ1[(µ, 1); (γ(µα− β), γα) ;−r2xγα] dx (1.13)(r, α, β, γ ∈ R+; γ(µα− β) > 1);S˜µ+1(r) =√pi(2r)µ−12Γ(µ+ 1)∞∫0xµ+12ex + 1Jµ− 12(rx)dx (r, µ ∈ R+); (1.14)S˜(α,0)µ (r; {k2/α}∞k=1) =2√pi(2r)µ−12Γ(µ)∞∫0xµ−12ex + 1Jµ− 12(rx)dx (r ∈ R+, µ > 12).(1.15)2. Mittag-Leﬄer type functionsThe special functions of the formEρ(z) =∞∑k=0zkΓ(ρk + 1), Eρ,µ(z) =∞∑k=0zkΓ(ρk + µ)(2.1)with complex ρ, µ ∈ C (Re(ρ) > 0, Re(µ) > 0) are known as Mittag-Leﬄer(M-L) functions.The former was introduced by Mittag-Leﬄer [13] and named after himin [7], vol. 3; and the latter is due to Agarwal [1] and also studied in detailsby Dzrbashjan (see e.g. [5]) who denoted it as Φ1/ρ(z;µ).Prabhakar [16] introduced the M-L type function Eδρ,µ(z) of the formEδρ,µ(z) =∞∑k=0(δ)kΓ(ρk + µ)zkk!(ρ, µ, γ ∈ C, Re(ρ) > 0), (2.2)where (δ)k is the Pochhammer symbol:(δ)0 = 1, (δ)k = δ(δ + 1) . . . (δ + k − 1) (k = 1, 2, . . . ).For δ = 1, we haveE 1ρ,µ(z) = Eρ,µ(z), E1ρ,1(z) = Eρ(z). (2.3)The function Eδρ,µ can be represented via the Fox-Wright Psi-function 1Ψ1and the generalized hypergeometric function 1Fm as follows:INTEGRAL REPRESENTATIONS OF . . . 131Eδρ,µ(z) =1Γ(δ)1Ψ1[(δ, 1), (µ, ρ); z], (2.4)Eδm,ν(z) =1Γ(δ)1Fm(δ;4(m; ν); zmm) (m ∈ N, δ, ν ∈ C). (2.5)Let m > 1 be an integer, ρ1, ρ2, . . . , ρm > 0 and µ1, µ2, . . . , µm be arbi-trary real numbers. By means of the ”multi-indices” (ρi), (µi), Kiryakova[9] introduced the so-called multi-index (m−tuple, multiple) M-L functions:E(1/ρi),(µi)(z) =∞∑k=0zkΓ(µ1 + k/ρ1)Γ(µ2 + k/ρ2) . . .Γ(µm + k/ρm). (2.6)About generalized fractional calculus differential and integral opera-tors and Laplace type integral transform, closely related to the functionsE(1/ρi),(µi), see also Al-Mussalam, Kiryakova and Vu Kim Tuan [2].The special case m = 2 of (2.6) was introduced by Dzrbashjan [5]. Hedenoted it byE(1/ρ1,1/ρ2),(µ1,µ2) = Φρ1,ρ2(z;µ1, µ2) =∞∑k=0zkΓ(µ1 + k/ρ1)Γ(µ2 + k/ρ2).(2.7)We mention here some special cases of (2.6), presented by Kiryakova in [9]:E1/ρ,µ(z) = E(1/ρ,0),(µ,1)(z) = Φp,∞(z;µ, 1), (2.8)Jν(z) =(z2)νE(1,1),(ν+1,1)(−z24)=(z2)νΦ1,1(−z24; 1, ν + 1), (2.9)E(1,1,...1),(µi+1)(z) = 1Ψm[(1, 1)(µi, 1)m1∣∣∣∣ z]=(m∏i=1Γ(µi))−1· 1Fm(1;µ1, µ2, . . . , µm; z). (2.10)Using the relations (2.4) and (2.5) we give here some integral representa-tions for the Mathieu type series S(α,β)µ (r,a) and S˜(α,β)µ (r,a) via (M-L) typefunctions:S(α,β)µ (r; {kγ}∞k=1) = 2∞∫0xγ(µα−β)−1ex − 1 Eµγα,γ(µα−β)(−r2xγα)dx; (2.11)132 Zˇ. TomovskiS˜(α,β)µ (r; {kγ}∞k=1) = 2∞∫0xγ(µα−β)−1ex + 1Eµγα,γ(µα−β)(−r2xγα)dx(r, α, β, γ ∈ R+; γ(µα− β) > 1); (2.12)S(α,β)µ (r; {kq/α}∞k=1) = 2∞∫0xq[µ−β/α]−1ex − 1 Eµq,q(µ−β/α](−r2xq)dx(r, α, β ∈ R+; µ− β/α > q−1; q ∈ N). (2.13)From the relation (see [16])L[tµ−1Eδρ,µ(wtρ)](s) = s−µ(1− ws−ρ)−δs ∈ C, Res > 0, |w/sρ| < 1, (2.14)and (1.11), upon setting f(t) = tq[µ−β/α]−2Eµq,q[µ−β/α](−r2tq), it easily fol-lows the integral representationS˜(α,β)µ (r; {kq/α}∞k=1) = 2∞∫0xq[µ−β/α]−1ex + 1Eµq,q[µ−β/α](−r2xq)dx. (2.15)Next, using the relation (2.7), we obtain also the integral representa-tions:S(α,α/2)µ+1 (r; {k2/α}∞k=1) = S(2,1)µ+1 (r; {k}∞k=1) = Sµ+1(r)=√pi22µΓ(µ+ 1)∫ ∞0x2µex − 1E(1,1),(µ+ 12 ,1)(−r2x24)dx; (2.16)S˜(α,α/2)µ+1 (r; {k2/α}∞k=1) =√pi22µΓ(µ+ 1)∫ ∞0x2µex + 1E(1,1),(µ+ 12,1)(−r2x24)dx(r, µ ∈ R+); (2.17)S(α,0)µ (r; {k2/α}∞k=1) =√pi22µ−1Γ(µ)∫ ∞0x2µ−1ex − 1E(1,1),(µ+ 12 ,1)(−r2x24)dx;(2.18)S˜(α,0)µ (r; {k2/α}∞k=1) =√pi22µ−1Γ(µ)∫ ∞0x2µ−1ex + 1E(1,1),(µ+ 12,1)(−r2x24)dx(r,∈ R+, µ > 12). (2.19)INTEGRAL REPRESENTATIONS OF . . . 1333. Mathieu-type series whose terms containmulti-index M-L functionsIn the recent articles ([14], [15], [22]) Pogany, Srivastava and Tomovskiconsidered special kind of Matieu-type series and their alternating variantswhose terms contain the Gauss hypergeometric function 2F1, generalized hy-pergeometric pFq and the Meijer G functions. The derived results concern,among others, closed integral form expressions for the considered series andbilateral bounding inequalities. Here we are interested in giving the integralexpression for the Mathieu-type series (and its alternating variants) whoseterms contain the multi-index M-L functions from [9] and [2].Definition. By a Fox’s H−function we mean a generalized hyper-geometric function, defined by means of the Mellin-Barnes-type contourintegralHm,np,q =[σ∣∣∣∣ (ak, Ak)p1(bk, Bk)q1]=12pii∫L′m∏k=1Γ(bk−Bks)n∏j=1(1− aj+sAj)q∏k=m+1Γ(1−bk+sBk)p∏j=n+1(aj−sAj)σsds,(3.1)where L′ is a suitable contour in C, the orders (m,n, p, q) are integers,0 ≤ m ≤ q, 0 ≤ n ≤ p and the parameters aj ∈ R, Aj > 0, j = 1, 2, . . . , p,bk ∈ R, Bk > 0, k = 1, 2, . . . , q are such that Aj(bk+l) 6= Bk(aj−l′−1), l, l′ =0, 1, 2, . . . .For various type of contours and conditions for existence and analyticityof function (3.1) in disks ⊂ C whose radii are ρ =p∏j=1A−Ajjq∏k=1BBkk > 0,one can see, for example [10], Appendix.We define the Mathieu-type series µ(1/ρi),(µi) and its alternating variantµ˜(1/ρi),(µi) as follows:µ(1/ρi),(µi)(a; r) =∞∑n=1E(1/ρi),(µi)(− r2an)aλn(an + r2)η, (3.2)µ˜(1/ρi),(µi)(a; r) =∞∑n=1(−1)n−1E(1/ρi),(µi)(− r2an)aλn(an + r2)η(λ, η, r ∈ R+), (3.3)where we make the convention that the real sequence a = {an}n∈N increasesand tends to ∞.We will need the Laplace transform of xλ−1H1,00,m+1(wx|·). Using thedefinition (3.1), for real w it follows easily134 Zˇ. Tomovski∞∫0e−Axxλ−1H1,00,m+1(wxρ∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)dx = A−λE(1/ρi),(µi)( wAp),where m > 0, Re{A}, Re{λ}+ ρ, min1≤j≤mRe{1− µj1/ρj}> 0. (3.4)Theorem. Let λ > 0, η > 0, r > 0 and let the real sequence a increaseand tend to ∞. Then,µ(1/ρi),(µi)(a; r) = ηI(λ, η + 1) + I(λ+ 1, η), (3.5)µ˜(1/ρi),(µi)(a; r) = ηI˜(λ, η + 1) + I˜(λ+ 1, η), (3.6)whereI(λ, η) =∞∫a1E(1/ρi),(µi)(−r2x)[a−1(x)]xλ(r2 + x)ηdx, (3.7)I˜(λ, η) =∞∫a1E(1/ρi),(µi)(−r2x)sin2(pi2 [a−1(x)]xλ(r2 + x)ηdx, (3.8)and a : R+ 7→ R+ is an increasing function such that a(x)|x∈N = a,a−1(x) denotes the inverse of a(x), [a−1(x)] stands for the integer part ofthe quantity a−1(x).P r o o f. Taking ξ = an + r2 in the formulaΓ(η)ξ−η =∞∫0e−ξttη−1dt (Re ξ > 0 , Re η > 0)and A = an, ρ = 1, w = −r2 in (3.4), we getµ(1/ρi),(µi)(a; r) =1Γ(η)∞∫0∞∫0H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)Da(s+ t)× e−r2stλ−1sη−1dsdt,whereDa(s+ t) =∞∑n=1ean(s+t) = (s+ t)∞∫a1e−(s+t)x[a−1(x)]dxis the integral representation of the Dirichlet series (see [8], [14], [15]).Hence,INTEGRAL REPRESENTATIONS OF . . . 135µ(1/ρi),(µi)(a; r) =1Γ(η)∞∫0∞∫0∞∫a1H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)× e−(r2+x)s−txsη−1tλ−1(s+ t)[a−1(x)]dsdxdt = Is + It,whereIs =1Γ(η)∞∫0∞∫0∞∫a1H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)× e−(r2+x)s−txtλ−1sη[a−1(x)]dsdxdt=1Γ(η)∞∫0∞∫a1H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)× e−txtλ−1[a−1(x)] ∞∫0e−(r2+x)ssηds dxdt=η∞∫a1[a−1(x)](r2 + x)η+1∞∫0H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1−µi, 1ρi )mi=1)tλ−1e−txdt dx= η∞∫a1[a−1(x)](r2 + x)η+1xλE(1/ρi),(µi)(−r2x)dx;It =1Γ(η)∞∫0∞∫0∞∫a1H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)× e−(r2+x)s−txsη−1tλ[a−1(x)]dsdxdt=1Γ(η)∞∫0∞∫a1e−(r2+x)ssη−1[a−1(x)]× ∞∫0H1,00,m+1(−r2t∣∣∣∣∣(0, 1), (1− µi, 1ρi )mi=1)tλe−txdt dsdx136 Zˇ. Tomovski=1Γ(η)∞∫a1E(1/ρi),(µi)(−r2x)[a−1(x)]xλ+1 ∞∫0e−(r2+x)ssη−1ds dx=∞∫a1E(1/ρi),(µi)(−r2x)[a−1(x)]xλ+1(r2 + x)ηdx.The derivation of (3.6) is similar to the previous proof; as the alternatingDirichlet series D˜a(x) equals to (see [14]):D˜a(x) =∞∑n=1(−1)n−1e−anx = x∞∫a1e−xtA˜(t)dt,where the so-called continuing function A˜(t) has been found easily in thefollowing manner:A˜(t) =∑n:an≤t(−1)n−1 = 1− (−1)[a−1(t)]2= sin2(pi2[a−1(t)]).Hence D˜a(s+ t) = (s+ t)∞∫a1e−x(s+t) sin2(pi2 [a−1(x)])dx, i.e. we get (3.6).References[1] R. P. Agarwal, A propos d’une note de M. Pierre Humbert. C. R.Acad. Sci. Paris 236 (1953), 2031-2032.[2] F. Al-Mussalam, V. Kiryakova, Vu Kim Tuan, A multi index Borel-Dzrbashjan transform. Rocky Mount. J. Math. 32, No 2 (2002), 409-428.[3] P. Cerone, C.T. Lenard, On integral forms of generalized Mathieuseries. J. Inequal. Pure Appl. Math. 4, No 5 (2003), 1-11; Article 100(electronic).[4] P.H. Diananda, Some inequalities related to an inaquality of Mathieu.Math. Ann. 250 (1980), 95-98.[5] M.M. Dzrbashjan, On the integral transformations generated by thegeneralized Mittag-Leﬄer function (in Russian). Izv. AN Arm. SSR13, No 3 (1960), 21-63.[6] D. Emersleben, Uber die Reihe∞∑k=1k/(k2 + c2)2. Math. Ann. 125(1952), 165-171.INTEGRAL REPRESENTATIONS OF . . . 137[7] A. Erde´lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (Ed-s),Higher Transcedental Functions, 3. Mc Graw-Hill Comp., New York(1955) (Reprinted by Krieger, Melbourne-Florida (1981)).[8] J. Karamata, Teorija i Praksa Stiltjesova Integrala (Theory and Ap-plication of the Stieltjes Integral). Srpska Akademija Nauka, Posebnaizdanja, Knjiga 144, Matematicˇki Institut, Knjiga 1, Beograd (1949).[9] V.S. Kiryakova, Multiple (multiindex) Mittag-Leﬄer functions and re-lations to generalized fractional calculus. J. Comput. Appl. Math. 118(2000), 241-259.[10] V. Kiryakova, Generalized Fractional Calculus and Applications. Ser.Research Notes in Math. Series 301, Pitman Longman, Harlow & Wi-ley, New York (1994).[11] E´.L. Mathieu, Traite´ de Physique Mathematique, VI-VII: The´ory del’E´lasticite´ des Corps Solides (Part 2). Gauthier-Villars, Paris (1890).[12] G. V. Milovanovic´, Some classes of orthogonal polynomials and appli-cations. Sci. Rev. 14 (1995), 5-28.[13] G. Mittag-Leﬄer, Sur la nouvelle fonction Eα(x). C. R. Acad. Sci.Paris 137 (1903), 554-558.[14] T.K. Pogany, H.M. Srivastava, Zˇ. Tomovski, Some families of Mathieua-series and alternating Mathieu a-series. Appl. Math. Comput. 173(2006), 69-108.[15] T.K. Pogany, Zˇ. Tomovski, On Mathieu-type series whose termscontain a generalized hypergeometric function pFq and Meijer’sG−function. Mathematical and Computer Modelling, In press.[16] T.R. Prabhakar, A singular integral equation with a generalizedMittag-Leﬄer function in the kernel. Yokohama Math. J. 19 (1971),7-15.[17] F. Qi, An integral expression and some inequalities of Mathieu typeseries. Rostock. Math. Kolloq. 58 (2004), 37-46.[18] H.M. Srivastava, Zˇ. Tomovski, Some problems and solutions involv-ing Mathieu’s series and its generalization. J. Inequal. Pure and Appl.Math. 5, No 2 (2004), 1-13; Article 45 (electronic).[19] H.M. Srivastava and H.L. Manocha, A Treatise on generating Func-tions. Halsted Press (Ellis Horwood Limited, Chichester), John Wileyand Sons, New York, Chichester, Brisbane and Toronto (1984).[20] Zˇ. Tomovski, New double inequalities for Mathieu type series. Univ.Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 15 (2004), 79-83.138 Zˇ. Tomovski[21] Zˇ. Tomovski, K. Trencˇevski, On an open problem of Bai-Ni Guo andFeng Qi. J. Inequal. Pure Appl. Math. 4, No 2 (2003), 1-7; Article 29(electronic).[22] Zˇ. Tomovski, T.K. Pogany, Integral expressions for Mathieu-type seriesof Fox’s H-function terms and related consequences. In: New Trendsin Mathematics and Informatics (Jubilee International Conference, 60Years Institute of Mathematics and Informatics, Bulgarian Academyof Sciences, July 6-8, 2007), Sofia.University ”St. Cyril and Methodius” Received: July 24, 2007Faculty of Natural Sciences and MathematicsInstitute of Mathematics, P.O. Box 162Skopje - 1000, Republic of MACEDONIAe-mail: tomovski@iunona.pmf.ukim.edu.mk

Integral Representations of Generalized Mathieu Series Via Mittag-Leffler Type Functions

The main purpose of this paper is to present a number of potentially useful integral representations for the generalized Mathieu series as well as for its alternating versions via Mittag-Leffler type functions

Zivorad Tomovski

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INTEGRAL REPRESENTATIONS OF GENERALIZED MATHIEU SERIES VIA MITTAG-LEFFLER TYPE FUNCTIONS

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Integral Representations of Generalized Mathieu Series Via Mittag-Leffler Type Functions

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