Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20By using integral representations for several Mathieu type series, a number
 of integral transforms of Hankel type are derived here for general families
of Mathieu type series. These results generalize the corresponding ones on
the Fourier transforms of Mathieu type series, obtained recently by Elezovic
et al. [4], Tomovski [19] and Tomovski and Vu Kim Tuan [20]

Tomovski, Živorad

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

ON HANKEL TRANSFORM OF GENERALIZEDMATHIEU SERIESZˇivorad TomovskiAbstractBy using integral representations for several Mathieu type series, a num-ber of integral transforms of Hankel type are derived here for general familiesof Mathieu type series. These results generalize the corresponding ones onthe Fourier transforms of Mathieu type series, obtained recently by Elezovicet al. [4], Tomovski [19] and Tomovski and Vu Kim Tuan [20].Mathematics Subject Classification: Primary 33E20, 44A10; Secondary33C10, 33C20, 44A20Key Words and Phrases: integral representations, Mathieu series, Han-kel transform, Fourier transform, Bessel function, Fox-Wright function, Sonine-Schafheitlin formula, Fox H-function1. Introduction and preliminariesThe following familiar infinite seriesS (r) =∞∑n=12n(n2 + r2)2(r ∈ R+) (1.1)is named after Emile Leonard Mathieu (1835-1890), who investigated it inhis 1890 work [8] on elasticity of solid bodies. An integral representation of(1.1) is given by (see [5])S (r) =1r∞∫0t sin (rt)et − 1 dt. (1.2)98 Zˇ. TomovskiSeveral interesting results dealing with integral representations and boundsfor a slight generalization of the Mathieu series with a fractional power,defined asSµ (r) =∞∑n=12n(n2 + r2)µ(r ∈ R+;µ > 1) , (1.3)can be found in the works by Diananda [2], Tomovski and Trencevski [16],Cerone and Lenard [1] and Tomovski [18]. Motivated essentially by theworks of Cerone and Lenard [1] (and Qi [13]), Srivastava and Tomovski [14]defined a family of generalized Mathieu seriesS(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 it is tacitly assumed that the positive sequencea = {an}∞n=1 = {a1, a2, a3, .....}(lim ann→∞=∞)is so chosen that the infinite series in definition (1.4) converges, that is, thatthe following auxiliary series ∞∑n=11aµα−βnis convergent. Comparing the definitions (1.1), (1.3) and (1.4), we see thatS2 (r) = S (r) and Sµ (r) = S(2,1)µ (r, {k}). Furthermore, the special casesS(2,1)2 (r; {ak}) , S(2,1)µ (r; {kγ}) and S(α,α/2)µ (r; {k}) have been investigatedby Qi [13], Diananda [2], Tomovski [17] and Cerone-Lenard [1].2. Some definitions and formulasIn this section we give some definitions and formulas, needed for com-putation of the Hankel transforms of S (r) , Sµ+1 (r) , S(α,β)µ(r;{k2/α})andS(α,β)µ (r; {kγ}).2.1. In order to evaluate the Hankel transform of S (r), we first setν = 12 in the Sonine-Schafheitlin formula (see for example, [6, p.692]):∞∫0t−λJµ (at) Jν (bt) dt =aµΓ ((µ+ ν − λ+ 1) /2)2λbµ−λ+1Γ (µ+ 1)Γ ((λ− µ+ ν + 1) /2)ON HANKEL TRANSFORM OF GENERALIZED . . . 99× 2F1(µ+ ν − λ+ 12,µ− ν − λ+ 12;µ+ 1;a2b2)(2.1)(< (µ+ ν + 1) > < (λ) > −1; 0 < a < b)with a corresponding expression for the case when 0 < b < a, which isobtained from (2.1) by interchanging a and b, and also µ and ν. In view ofthe relationshipJ1/2 (x) =√2pixsinx , (2.2)we find from the Sonine-Schafheitlin formula (2.1) that∞∫0t−λ−1/2Jµ (at) sin (bt) dt =√pib2aµΓ[12 (µ− λ+ 3/2)]2λbµ−λ+1Γ (µ+ 1)Γ[12 (λ− µ+ 3/2)]× 2F1(µ− λ+ 3/22,µ− λ+ 1/22;µ+ 1;a2b2)(2.3)(<(µ+ 32) > < (λ) > −1; 0 < a < b),together with the corresponding integral derived from the analogue of (2.1)for the case when 0 < b < a. Thus, by further applying integral formula(2.3) and its companion when 0 < b < a, we obtain (with λ = 0, µ→ ν):∞∫0r−1/2 sin (rt) Jν (rx) dr =√pi2{Ω(1)ν (x; t) : 0 < x < tΩ(2)ν (x; t) : 0 < t < x(< (ν) > −12),(2.4)whereΩ(1)ν (x; t) =xνtν+12Γ(ν2 +34)Γ (ν + 1)Γ(34 − ν2) 2F1(ν2 + 34 , ν2 + 14; ν + 1; x2t2)(0 < x < t)Ω(2)ν (x; t) =tx32Γ(ν2 +34)Γ(32)Γ(ν2 +14) 2F1(ν2 + 34 , 34 − ν2 ; 32; t2x2)(0 < t < x) (<(ν) > −12).2.2. If we apply the Sonine-Schafheitlin formula (2.1), first with λ =µ − 1, µ → µ − 1/2, a = t, b = x, (0 < a < b), and then with λ = µ − 1,µ→ ν, ν → µ− 12 , a = x, b = t (0 < b < a), we get100 Zˇ. Tomovski∞∫0r1−µJµ−1/2 (rt) Jν (rx) dr ={Θ(1)ν (µ;x, t) : 0 < t < xΘ(2)ν (µ;x, t) : 0 < x < t(< (µ) > 0, < (ν) > −32), (2.5)whereΘ(1)ν (µ;x, t) =tµ−12Γ(ν2 +34)2µ−1x32Γ (µ+ 1/2) Γ(ν2 +14) 2F1(34 + ν2 , 34 − ν2 ;µ+ 12; t2x2)(0 < t < x, < (µ) > 0, < (ν) > −32), (2.6)Θ(2)ν (µ;x, t) =xνΓ(ν2 +34)2µ−1tν−µ+2Γ (ν + 1)Γ(µ− ν2 − 14)× 2F1(34+ν2,ν2− µ+ 54; ν + 1;x2t2)(0 < x < t, < (µ) > 0, < (ν) > −32). (2.7)2.3. For the evaluation of the Hankel transform of the general Mathieuseries S(α,β)µ(r;{k2/α}), we need the definition of the Meijer G-function andthe Fox H-function (for more details, see e.g. in [3, Vol.1], [12]).Definition 1. 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∣∣∣∣ (ap, αp)(bq, βq)]=12pii∫Lm∏k=1Γ (bk − sβk)n∏j=1Γ (1− aj + sαj)q∏k=m+1Γ (1− bk + sβk)p∏j=n+1Γ (aj − sαj)σsds,(2.8)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, αj > 0, j = 1, 2, .., p,bk ∈ R, βk > 0, k = 1, 2, .., q are such that αj (bk + l) 6= βk (aj − l′ − 1),l, l′ = 0, 1, 2, ...Many special functions are particular cases of the H-function. For ex-ample, if αl = βj = 1 (l = 1, 2, .., p; j = 1, 2, ..., q), it reduces to the MeijerG-function Gm,np,q (σ) (for definition and properties, see [3, Vol.1]):ON HANKEL TRANSFORM OF GENERALIZED . . . 101Hm,np,q[σ∣∣∣∣ (ap, αp)(bq, βq)]= Gm,np,q[σ∣∣∣∣ (ap)(bq)]. (2.9)On the other hand, the Fox-Wright Ψ-function is the following specialcase of the H-function, see for example in [15]:pΨq [(ap, αp) ; (bq, βq) ;x] = H1,pp,q+1[−x∣∣∣∣ (1− ap, αp)(0, 1), (1− bq, βq)]. (2.10)Next we use of the following Mellin transform of a product of two hy-pergeometric functions, proven by Miller and Srivastava [9] (see also [12,Sect.2.22, p.333])F (s) =∞∫0rs−10F1(−; 1 + µ;−a2r2) 1F2 (α;β, 1 + ν;−b2r2) dr=12asΓ (1 + µ) Γ (1 + ν) Γ (β)Γ (α)G1,23,3[b2a2∣∣∣∣ 1− s/2, 1− α, 1 + µ− s/20,−ν, 1− β](2.11)(a > 0, b > 0, 0 < <(s) < <(2 + µ+ ν + β − α), 0 < <(s) < <(32+ 2α+ µ)).Substituting the relation (see [11, p.727])0F1(−; 1 + µ;−a2r2) = Γ (1 + µ) Jµ (2ar)(ar)µ(2.12)into the integral formula (2.11) with µ→ ν, s = µ+ 32 → ν + 32 , a→ x2 , α→µ, β → µ− βα , ν = µ− βα − 12 , b = t2 , we get∞∫0r1/2Jν (rx)1 F2(µ;µ− βα, µ− βα+12;−r2t24)dr =√2x3/2×Γ(µ− βα + 12)Γ(µ− βα)Γ (µ)G1,23,3[t2x2∣∣∣∣ 1/4− µ/2, 1− µ, 1/4 + ν − µ/20,−µ+ β/α+ 1/2, 1− µ+ ηα](2.13)(<(ν) > −32, <(µ) > 0, <(µ− 2βα) > 0, t > 0, x > 0).2.4. In order to evaluate the Hankel transform of S(α,β)µ (r; {kγ}), weapply the known integral formula from [11, p.355] with p = 1, q = 2, r =2, α = 3/2:102 Zˇ. Tomovski∞∫0x1/2Jν (σx) H1,11,2[ωx2∣∣∣∣ (1− µ, 1)(0, 1), (1− γ(µα− β), γα)]dx=√2σ3/2H1,23,2[4ωσ2∣∣∣∣ (1/4− ν/2, 1), (1− µ, 1), (1/4 + ν/2, 1)(0, 1), (1− γ(µα− β), γα)](2.14)(ω ∈ R, σ ∈ R+, < (µ) > 0, < (ν) > −32).Using relation (2.10) with p = 1, q = 1, by (2.14) we get the followingintegral formula:∞∫0x1/2Jν (σx)1Ψ1[(µ, 1) ; (γ (µα− β) , γα) ;−x2tγα] dx=√2σ3/2H1,23,2[4tγασ2∣∣∣∣ (1/4− ν/2, 1), (1− µ, 1), (1/4 + ν/2, 1)(0, 1), (1− γ(µα− β), γα)](2.15)(t ∈ R+, σ ∈ R+, < (µ) > 0,< (ν) > −32).3. Evaluation of the Hankel transformThe Hankel transform of order ν is defined byHν (f (r)) (x) =∞∫0f (r) Jν (rx)√rx dr, (3.1)where Jν is the Bessel function of the first kind of order ν with < (ν) ≥ −12 .In view of the relationship (2.1) andJ−1/2 (x) =√2pixcosx, (3.2)the Hankel transform reduces to the sin-Fourier and cos-Fourier transforms:(Fs f) (x) =√2pi∞∫0sin (rx) f (r) dr, (Fc f) (x) =√2pi∞∫0cos (rx) f (r) dr.For a generalization of the Hankel type transform as well as of the sin-Fourierand cos-Fourier transforms, see Luchko and Kiryakova [7].ON HANKEL TRANSFORM OF GENERALIZED . . . 1033.1. Using the integral formula (2.4), we first evaluate the Hankel trans-form of S (r):Hν (S (r)) (x)=∞∫0S (r) Jν (rx)√rxdr=√x∞∫01√r ∞∫0t sin (rt)et − 1 dtJν (rx) dr=√x∞∫0tet − 1 ∞∫0r−1/2 sin (rt) Jν (rx) dr dt=√pix2 x∫0tet − 1Ω(2)ν (x; t) dt+∞∫xtet − 1Ω(1)ν (x; t) dt . (3.3)Using the relation [6, p.1041]2F1(1,12;32; z2)=12zln1 + z1− z ,we getΩ(1)1/2 (x; t) =√xtΓ(32)Γ(12) 2F1(1, 12; 32; x2t2)=1pi√xlnt+ xt− x (0 < x < t)Ω(2)1/2 (x; t) =tx√xΓ(32)Γ(12) 2F1(1, 12; 32; t2x2)=1pi√xlnx+ tx− t (0 < t < x) .HenceH1/2 (S (r)) (x) = (Fs S (r) ) (x)=√pix2 x∫0tet − 11pi√xlnx+ tx− tdt+∞∫xtet − 11pi√xlnt+ xt− xdt=1√2pi x∫0tet − 1 lnx+ tx− tdt+∞∫xtet − 1 lnt+ xt− xdt=1√2piPV ∞∫0tet − 1 ln∣∣∣∣ t+ xt− x∣∣∣∣ dt (x > 0) , (3.4)where the Cauchy Principal Value (PV) of the last integral is assumed toexist. By a direct computational, the same formula (3.4) was recently provedby Elezovic et al [4].104 Zˇ. Tomovski3.2. The integral representation of Sµ+1 (r), obtained by Cerone andLenard in [1], is given bySµ+1 (r) =√pi(2r)µ−1/2 Γ (µ+ 1)∞∫0tµ+12et − 1Jµ−1/2 (rt) dt(r, µ ∈ R+) . (3.5)Applying integral formula (2.5), we obtainHν (Sµ+1 (r)) (x) =∞∫0Sµ+1 (r) Jν (rx)√rxdr=√pix2µ−1/2Γ (µ+ 1)∞∫0r1−µ ∞∫0tµ+12et − 1Jµ−1/2 (rt) dtJν (rx) dr=√pix2µ−1/2Γ (µ+ 1)∞∫0tµ+12et − 1 ∞∫0r1−µJµ−1/2 (rt)Jν (rx) dr dt=√pix2µ−1/2Γ (µ+ 1) x∫0tµ+12et − 1Θ(1)ν (µ;x, t) dt+∞∫xtµ+12et − 1Θ(2)ν (µ;x, t) dt(3.6)(µ ∈ R+, <(ν) > −12, x > 0).3.3. In order to evaluate the Hankel transform of S(α,β)µ(r;{k2/α}), weapply its integral representation from [14]:S(α,β)µ(r;{k2/α})=2Γ(2[µ− βα]) ∞∫0x2(µ−βα)−1ex − 1× 1F2(µ;µ− βα, µ− βα+12;−r2x24)dx (3.7)(r, α, β ∈ R+, µ− βα>12)and integral formula (2.13). Thus, we haveHν(S(α,β)µ(r;{k2/α}))(x) =∞∫0S(α,β)µ(r;{k2/α})Jν (rx)√rxdrON HANKEL TRANSFORM OF GENERALIZED . . . 105=2√xΓ(2[µ− βα]) ∞∫0t2(µ−βα)−1et − 1× ∞∫0r1/21F2(µ;µ− βα, µ− βα+12;−r2t24)Jν (rx) dr dt=2√2pi22(µ−βα)−1xΓ (µ) ∞∫0t2(µ−βα)−1et − 1 G1,23,3(t2x2∣∣∣∣ 14−µ2 ,1−µ, 14+ν−µ20,−µ+ βα+ 12,1−µ+ βα)dt(< (ν) > −12, µ− 2βα> 0, µ− βα>12, x > 0). (3.8)3.4. The integral representation of S(α,β)µ (r; {kγ}∞k=1) obtained by Sri-vastava and Tomovski [14] is given byS(α,β)µ (r; {kγ}∞k=1) =2Γ (µ)∞∫0xγ(µα−β)−1ex − 1× 1Ψ1[(µ, 1) ; (γ (µα−β) , γα) ;−r2xγα] dx (3.9)(r, α, β, γ ∈ R+, γ (µα−β) > 1) .Using this integral representation and (2.15), we getHν(S(α,β)µ (r; {kγ}))(x) =∞∫0S(α,β)µ (r; {kγ}) Jν (rx)√rxdr =2√xΓ (µ)×∞∫0tγ(µα−β)−1et − 1 ∞∫0r1/2Jν (rx)Ψ1[(µ, 1) ; (γ (µα− β) , γα) ;−r2tγα] dr dt=2√2xΓ (µ)∞∫0tγ(µα−β)−1et − 1 H1,23,2(4tγαx2∣∣∣∣( 14− ν2 ,1),(1−µ,1),( 14+ ν2 ,1)(0,1),(1−γ(µα−β),γα) ) dt(< (µ) > 0,< (ν) > −32, γ (µα− β) > 1, x > 0). (3.10)106 Zˇ. TomovskiReferences[1] P. Cerone, C.T. Lenard, On integral forms of generalized Mathieuseries. J. Inequal. Pure Appl. Math. 4, No 5 (2003), Article 100, 1-11(electronic).[2] P.H. Diananda, Some inequalities related to an inequality of Mathieu.Math. Ann. 250 (1980), 95-98.[3] A. Erde´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi (Ed-s),Higher Transcedental Functions, Vols. 1-2-3, Mc Graw-Hill, N. York -Toronto - London (1953).[4] N. Elezovic, H.M. Srivastava, Zˇ. Tomovski, Integral representationsand integral transforms of some families of Mathieu type series. IntegralTransforms and Special Functions 19, No 7 (2008), 481-495.[5] O. Emersleben, Uber die Reihe. Math. Ann. 125 (1952), 165-171.[6] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products.Academic Press, N. York - London (1965).[7] Yu.F. Luchko, V.S. Kiryakova, Generalized Hankel transforms forhyper-Bessel differential operators. C.R. Acad. Bulgare Sci. 53, No8 (2000), 17-20.[8] E.L. Mathieu, Traite´ de Physique Mathematique. VI-VII: Theory del’Elasticite´ des Corps Solides (Part 2). Gauthier-Villars, Paris (1890).[9] A.R. Miller and H.M. Srivastava, On the Mellin transform of a productof hypergeometric functions, J. Austral. Math. Soc. Ser. B40 (1998),222-237.[10] T.K. Pogany, H.M. Srivastava, Zˇ. Tomovski, Some families of Mathieua-series and alternating Mathieu a-series. Appl. Math. Computation173 (2006), 69-108.[11] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series,Vol. 1: Elementary Functions; Vol. 2: Special Functions. Gordon andBreach Sci. Publ., N. York (1992); Russian original: Nauka, Moscow(1981).ON HANKEL TRANSFORM OF GENERALIZED . . . 107[12] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Se-ries (More Special Functions). Gordon and Breach Sci. Publ., N. York(1990); Russian original: Nauka, Moscow (1981).[13] F. Qi, An integral expression and some inequalities of Mathieu typeseries. Rostock. Math. Kolloq. 58 (2004), 37-46.[14] H.M. Srivastava, Zˇ. Tomovski, Some problems and solutions involvingMathieu’s series and its generalizations. J. Inequal. Pure Appl. Math.5, No 2 (2004), Article 45, 1-13 (electronic).[15] H.M. Srivastava and H.L. Manocha, A Treatise of Generating Func-tions. Halsted Press (Ellis Horwood Ltd, Chichester), J. Wiley & Sons,N. York - Chichester - Brisbane -Toronto (1984).[16] Zˇ. Tomovski, K. Trencevski, On an open problem of Bai-Ni Guo andFeng Qi. J. Inequal. Pure Appl. Math. 4, No 2 (2003), Article 29, 1-7(electronic).[17] Zˇ. Tomovski, New double inequality for Mathieu series, Univ. BeogradPubl. Elektrotehn. Fak. Ser. Mat. 15 (2004), 79-83.[18] Zˇ. Tomovski, Integral representations of generalized Mathieu series viaMittag-Leﬄer type functions. Fract. Calc. and Appl. Anal. 10, No 2(2007), 127-138.[19] Zˇ. Tomovski, New integral and series representations of the generalizedMathieu series. Appl. Anal. Discrete Math. 2, No 2 (2008), 205-212.[20] Zˇ. Tomovski, Vu Kim Tuan, On Fourier transforms and summationformulas of generalized Mathieu series. Math. Science Res. J., To ap-pear (2009).Received: August 25, 2008University ”St. Cyril and Methodius”Faculty of Natural Sciences and MathematicsInstitute of MathematicsP.O. Box 162, 1000 – Skopje, Republic of MACEDONIAe-mail: tomovski@iunona.pmf.ukim.edu.mk

On Hankel Transform of Generalized Mathieu Series

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On Hankel Transform of Generalized Mathieu Series

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