Drascic Ban, Biserka

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 11 (2010), No 1, pp. 53-62 DOI: 10.18514/MMN.2010.248On multiple Mathieu (a, λ)seriesBiserka Dra²£i£ BanMiskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 11 (2010), No. 1, pp. 53–62ON MULTIPLE MATHIEU .a;/-SERIESBISERKA DRASˇCˇIC´ BANReceived 2 December, 2009Abstract. In this paper we introduce the multiple Mathieu .a;/-series. We obtain two integralrepresentations for multiple Mathieu .a;/-series applying Ivanov’s and then Poga´ny’s variant ofmultiple Euler-Maclaurin summation formula. Then, a bilateral bounding inequality is derivedby virtue of the achieved integral expressions. Finally, the special case of multiple Mathieu.a;/-series, the multiple Mathieu a-series has been investigated.2000 Mathematics Subject Classification: 26B40, 26D15, 33E20, 40B05Keywords: Mathieu series, multiple Mathieu series, multiple Mathieu .a;/-series, multipleMathieu a-series, Dirichlet series, r-dimensional Euler-Maclaurin summation formula, Gammafunction1. INTRODUCTIONThe generalization of the classical Mathieu series [3]S.r/D1XnD12n.n2C r2/2 .r > 0/ (1.1)has been introduced by Cerone and Lenard [1]:S.r;;˛;ˇ;a/D1XnD1a˛n.aˇn C r2/.r;;˛;ˇ;aD .an/ > 0/: (1.2)The series (1.2) is in the focus of interest by numerous authors, such as Poga´ny[5, 6], Qi [11, 12], Srivastava and Tomovski [13, 14]. However, according to ourbest knowledge, the only work on the multidimensional generalization of the series(1.2) is the paper [10] by Poga´ny and Tomovski, where they introduce a generalizedmultiple Mathieu series of the formSrp .s;q;/DXn2Nr2njsj.hnq ;nqiC/pC1 ;Supported by the Ministry of Sciences, Education and Sports of Croatia under the Research ProjectNo. 112-2352818-2814.c 2010 Miskolc University Press54 BISERKA DRASˇCˇIC´ BANwhere nq WD .nq11 ; : : : ; ;nqrr /, n˛jsjWD n˛1s11   n˛rsrr , s;q have positive coordinates,i. e., sl ;ql > 0, l D 1; : : : ; r , while ha;bi stands for inner product in Rr . For r D 1,the above series, obviously, reduces to the classical Mathieu series (1.1).Poga´ny and Tomovski found two integral representations of the multiple seriesSrp .s;q;/ (see Theorem 1, Eq. (9) and Theorem 3, Eq. (19) in [10]). They alsoderived a bilateral bounding inequality [10, Theorem 2] and established two otherbounds [10, Theorems 4 and 5].2. SUMMATION FORMULA FOR FINITE MULTIPLE SUMSThe well-known Euler–Maclaurin summation formula has the formlXnDkf .n/DZ lkf .x/dxC 12 f .k/Cf .l/CmXjD1B2j.2j /Šf .2j 1/.l/ f .2j 1/.k/ Z lkB2m.x/.2m/Šf .2m/.x/dx .m 2N/;(2.1)where Bp.x/ D .xCB/p, 0  x < 1, stands for the Bernoulli polynomial of orderp 2 N and Bk D Bk are the Bernoulli numbers. One can rewrite it in a condensedformlXnDkC1an DZ lk a.x/Cfxga0.x/dx  Z lkda.x/dx; (2.2)for a 2 C 1Œk; l, an D a.n/, k; l 2Z, k < l , whered WD 1Cfxg @@xand fxg stands for the fractional part of a real number x (see (3) in [9] and (6.5) in[8]).For the multidimensional bounded summation domain D, we use the summa-tion formulas derived by Mu¨ller [4], Ivanov [2], and another type of formula dueto Poga´ny [7].Now the role of the Bernoulli polynomialsBp.x/ in (2.1) is played by the so-calledbasic functions (“Grundfunktion,” see [4])G.x1; : : : ;xr/, which satisfy the followingconditions:(1) G is a 1-periodic function in all variables;(2) On the lattice Zr , the function G satisfies the equationGCG DXkW 42k2De2ihk;xi; (2.3)MULTIPLE MATHIEU .a;/-SERIES 55where  is the Laplace operator and the summation in (2.3) is carried outover all kD .k1; : : : ;kr/ 2Zr such that 42k2 D 42hk;ki D .Let us introduce the following notation: n D .n1; : : : ;nr/ 2 Nr0, where N0 Df0;1;2; : : :g; x D .x1; : : : ;xr/ 2 Rr , dx D dx1    dxr , L D L @@xis a linear dif-ferential operator with real constant coefficients, which is a polynomial in @@xj.Let us putH.x/DXnWL.2in/D0e2ihn;xi: (2.4)Let G be a basic function of the operator L if it has period 1 with respect to eachvariable and satisfies the equalityLG DXn2Zre2ihn;xi:It follows thatG.x/DXn2Zr0 e2ihn;xiL.2in/; (2.5)where 0 denotes the absence of the term with zero denominator in the sum.Let the boundary @D of D be smooth. Then, by Green’s formula, we haveZxD.uLv vMu/ dx DZ@DP.u;v/ds; (2.6)where xDDDC@D,M is a conjugate ofL, and P.u;v/ is a polynomial with respectto u and v and their partial derivatives.Theorem 1 ([2, Theorem 1]). Let us assume that the boundary @D ofD is smoothand does not contain integer points. Let L be a linear differential operator of orderp with constant coefficients and f 2 Cp. xD/, where xD DDC@D. ThenXn2Df .n/DZDf .x/H.x/dxCZ@DP.f;G/dsCZDGM.f /dx; (2.7)where M is a conjugate of L, whereas H and G are defined by (2.4) and (2.5).The following result of [7] is a multidimensional generalization of (2.2).Theorem 2 ([7]). Assume that a W RrC! RC is a function satisfying the conditionof differentiability@ra@x1   @xr 2 C rYjD1Œ0;nj ;and, for any j D .j1; : : : ;jr/with 0 jl  nl , l D 1;2; : : : ; r , put aj WD a.j1; : : : ;jr/.In what follows, the expression L.2in/ stands for the value of that polynomial where, instead of@@xj, one puts 2inj .56 BISERKA DRASˇCˇIC´ BANThenrXlD1nlXjlD0aj D a0CrXlD1Z nl0dl a.xl/dxlCX1j<krZ nj0Z nk0dj dk a.xj ;xk/dxj dxkC CZ n10Z n20  Z nr0d1d2   dr a.x1;x2; : : : ;xr/dx1 dx2    dxr ;(2.8)where a0  a.0; : : : ;0/, a.xj1 ; : : : ;xjk /D a.x/jxmD0; m2f1;:::;rgnfj1;:::;jkg, anddj WD 1Cfxj g @@xj:3. MULTIPLE MATHIEU .a;/-SERIESLet a;WRrC! RC, where r > 1, be certain functions, a and  be their restric-tions to Nr0, i. e., ajNr0 D .an/n2Nr0 D .a.n1; : : : ;nr//n2Nr0 ; jNr0 D .n/n2Nr0 D..n1; : : : ;nr//n2Nr0 , and  > 0 and  > 0 be parameters. Then the seriesS.a;I/ WDXn2Nr0an.nC/ (3.1)is called a multiple Mathieu .a;/-series.We are interested in deriving an integral representation of the series (3.1). Let usbegin with transforming S.a;I/ via the Gamma function. We haveS.a;I/D 1  ./Xn2Nr0anZ 10e .nC/xx 1 dxD 1  ./Z 10e xx 1 Xn2Nr0ane nx!dx:The inner r-fold Dirichlet seriesDa;.x/DXn2Nr0ane nx;by virtue of the Laplace integral formula, can be rewritten in the formDa;.x/D xZ 10e xt XnWntan.t/!dt D xZ 10e xtAn.t/dt:MULTIPLE MATHIEU .a;/-SERIES 57Putting all back into the expression for S.a;I/, we obtainS.a;I/D 1  ./Z 10e xxZ 10e xtAn.t/dtD 1  ./Z 10Z 10e .Ct/xx dxAn.t/dtD Z 10An.t/.C t /C1 dt:Now we have to calculate the sum An.t/. Let DD fn W n  t; t > 0g and let y1 Dy1.t/; : : : ;yr D yr.t/ 2N0 be such thatn D .y1; : : : ;yr/ t: (3.2)The summation domain is the r-dimensional parallelepipedDDQrjD1Œ0;yj  and,obviously, dim@D D r   1. Let us set dsr 1 D ds1    dsr 1. Let L be an ellipticdifferential operator, M be its conjugate, P be the polynomial defined by (2.6), andH and G be given by (2.4) and (2.5), respectively. By Ivanov’s formula (2.7), weobtainAn.t/DZDa.x/H.x/dxCZ@DP.a;G/dsr 1CZDGM.a/dx:Putting this expression for An.t/ back to S.a;I/, we immediately arrive at thefollowing statement.Theorem 3. Let a;WRrC ! RC, an D a.n1; : : : ;nr/, n D .n1; : : : ;nr/, n 2Nr0, ; > 0, and r 2 N2. Let t > 0 and let y1 D y1.t/; : : : ;yr D yr.t/ 2 N0 besuch that (3.2) holds. Then the multiple Mathieu .a;/-series admits the integralrepresentationS.a;I/D l 10ZDa.x/H.x/dxCZ@DP.a;G/dsr 1CZDGM.a/dx.C t /C1 dt;where DDQrjD1Œ0;yj .If we use formula (2.8) to calculate An.t/ in S.a;I/, we deduceTheorem 4. Let a;WRrC! RC, anD a.n1; : : : ;nr/, nD .n1; : : : ;nr/, n 2Nr0, > 0; > 0, r > 0, and y1 D y1.t/; : : : ;yr D yr.t/ 2N0 such thatn D .y1; : : : ;yr/ t:Then multiple Mathieu .a;/-series has the following integral representationS.a;I/D  a0CZ 10 rXlD1Z yl0dl a.xl/dxl58 BISERKA DRASˇCˇIC´ BANCX1j<krZ yj0Z yk0dj dk a.xj ;xk/dxj dxkC CZ y10Z y20  Z yr0d1   dr a.x1; : : : ;xr/dx1    dxrdt.C t /C1 :4. BILATERAL BOUNDING INEQUALITYWe estimate An.t/ given by Poga´ny’s formula (2.8). We will focus here on thecase where r D 2, since the higher order terms are far more complicated and theprocedure is identical to that applicable for r D 2.Let aWR2C! R be monotone in both variables. Since, for j D 1;2,dja 2ha;aC @a@xj.if a " in xj /I (4.1)dja 2aC @a@xj;ai.if a # in xj;successively applying the operators d1 and d2 in (2.8), we estimateA2 Dn1XjD0n2XkD0ajkD a0C2XjD1Z nj0dj a.xj /dxj CZ n10Z n20dj dk a.xj ;xk/dxj dxk :Assume that a ". ThenA2.t/ a0C2XjD1Z nj0 a.xj /Ca0.xj /dxjCZ n10Z n20a.x1;x2/C @a.x1;x2/@x1C @a.x1;x2/@x2C @2a.x1;x2/@x1@x2dx1 dx2 DWR2:MULTIPLE MATHIEU .a;/-SERIES 59Now we haveR2 D a0C2XjD1Z nj0a.xj /dxj Ca.nj / a0CZ n10Z n20a.x1;x2/dx1 dx2CZ n10 a.x1;n2/ a.x1;0/dx1CZ n20 a.n1;x2/ a.0;x2/dx2Ca.n1;n2/ a.n1;0/ a.0;n2/Ca0D a0C2XjD1Z nj0a.xj /dxj Ca.nj / a0CZ n10Z n20a.x1;x2/dx1 dx2CZ n10 a.x1;n2/ a.x1/dx1CZ n20 a.n1;x2/ a.x2/dx2Ca.n1;n2/ a.n1/ a.n2/Ca0D a.n1;n2/CZ n10a.x1;n2/dx1CZ n20a.n1;x2/dx2CZ n10Z n20a.x1;x2/dx1 dx2 :(4.2)Similarly, by virtue of (4.1), we haveA2  a0CZ n10a.x1/dx1CZ n20a.x2/dx2CCZ n10Z n20a.x1;x2/dx1 dx2 DWL2 :(4.3)This proves the following result.Theorem 5. Let aWR2C! R and a 2 C 2 Œ0;n1 Œ0;n2, monotone in both vari-ables. Then for the two-dimensional Euler–Maclaurin summation formula (2.8) thefollowing bilateral bounds are validL2 <n1XjD0n2XkD0ajk R2 ; .a "/;R2 n1XjD0n2XkD0ajk <L2 ; .a #/;where a.x/jN20 D aD .ajk/N20 and bounds L2, R2 are given in (4.3), (4.2).If we apply the above theorem to integral representation of multiple Mathieu.a;/-series given in Theorem 4, for r D 2 with simple exchange y1! n1, y2! n2we obtain60 BISERKA DRASˇCˇIC´ BANTheorem 6. Let a;WR2C! R, a; 2 C 2.Œ0;y1 Œ0;y2/, where yj D yj .t/ 2N0, j D 1;2, for some fixed t > 0, such that .y1;y2/ t and a be monotone in bothvariables. Then the following bilateral inequalities holdZ 10yL2.t/.C t /C1 dt < S.a;I/ Z 10yR2.t/.C t /C1 dt; .a "/;Z 10yR2.t/.C t /C1 dt S.a;I/ < Z 10yL2.t/.C t /C1 dt; .a #/;where yX2.t/DX2 y1;y2, X 2 fR;Lg, that isyR2.t/D a y1.t/;y2.t/CZ y1.t/0a x1;y2.t/dx1CZ y2.t/0a y1.t/;x2dx2CZ y1.t/0Z y2.t/0a.x1;x2/dx1 dx2yL2.t/D a0CZ y1.t/0a.x1/dx1CCZ y2.t/0a.x2/dx2CZ y1.t/0Z y2.t/0a.x1;x2/dx1 dx2:Remark 1. Since y1 and y2 are solutions of the Diophantine equation .y1;y2/t , they are not uniquely determined, nor are the functions yL2, yR2.5. MULTIPLE MATHIEU a-SERIESIf we take   a, it is straightforward to see that from multiple Mathieu .a;/-series we easily obtain the seriesS.aI/ WDXn2Nr0an.anC/which we call multiple Mathieu a-series. Now from integral representations for mul-tiple Mathieu .a;/-series we directly get the following results.Corollary 1 (of Theorem 3). Let aWRrC!RC, anD a.n1; : : : ;nr/, n2Nr0, ; >0, and r 2N2. Let t > 0 and ya1 D ya1.t/; : : : ; yar D yar.t/ 2N0 such thatan D a.ya1; : : : ; yar/ t:If DDQrjD1Œ0; yaj  then multiple Mathieu a-series have the following integral rep-resentationS.aI/D l 10ZDa.x/H.x/dxCZ@DP.a;G/dsr 1CZDGM.a/dx.C t /C1 dt;MULTIPLE MATHIEU .a;/-SERIES 61with L, H and G keeping the heretofore meaning.Corollary 2 (of Theorem 4). Let aWRrC!RC, anD a.n1; : : : ;nr/, n2Nr0, > 0, > 0, r > 0, and ya1 D ya1.t/; : : : ; yar D yar.t/ 2N0 such thatan D a.ya1; : : : ; yar/ t:Then multiple Mathieu a-series possesses the integral representationS.aI/D  a0CZ 10 rXlD1Z yal0dl a.xl/dxlCX1j<krZ yaj0Z yak0dj dk a.xj ;xk/dxj dxkC CZ Oa10  Z yar0d1   dr a.x1; : : : ;xr/dx1    dxr1.C t /C1 dt:Corollary 3 (of Theorem 5). Let aWR2C ! R, a 2 C 2 Œ0; ya1 Œ0; ya2,  > 0, > 0, and ya1; ya2 2N0 such thatan D a.ya1; ya2/ t:Then the following bilateral inequalities holdZ 10La2.t/.C t /C1 dt < S.aI/ Z 10Ra2 .t/.C t /C1 dt; .a "/;Z 10Ra2 .t/.C t /C1 dt  S.aI/ < Z 10La2.t/.C t /C1 dt; .a #/;whereLa2.t/D a0C2XjD1Z Oaj0a.xj /dxj CCZ ya10Z ya20a.x1;x2/dx1 dx2 ;Ra2 .t/D a.ya1; ya2/CZ ya10a.x1; ya2/dx1CZ ya20a.ya1;x2/dx2CZ ya10Z ya20a.x1;x2/dx1 dx2 :REFERENCES[1] P. Cerone and C. T. Lenard, “On integral forms of generalised Mathieu series,” JIPAM. J. Inequal.Pure Appl. Math., vol. 4, no. 5, pp. Article 100, 11 pp. (electronic), 2003.[2] V. K. Ivanov, “Higher-dimensional generalizations of the Euler summation formula,” Izv. Vyssˇ.Ucˇebn. Zaved. Matematika, vol. 1963, no. 6 (37), pp. 72–80, 1963.[3] E. Mathieu, Traite´ de Physique Mathe´matique, VI-VII: The´orie de l’e´lasticitre´ des corps solides.Paris: Gauthier-Villars, 1890.62 BISERKA DRASˇCˇIC´ BAN[4] C. Mu¨ller, “Eine Verallgemeinerung der Eulerschen Summenformel und ihre Anwendung aufFragen der analytischen Zahlentheorie,” Abh. Math. Sem. Univ. Hamburg, vol. 19, no. 1-2, pp.41–62, 1954.[5] T. K. Poga´ny, “Integral representation of a series which includes the Mathieu a-seri-es,” J. Math. Anal. Appl., vol. 296, no. 1, pp. 309–313, 2004. [Online]. Available:http://dx.doi.org/10.1016/j.jmaa.2004.04.020[6] T. K. Poga´ny, “Integral representation of Mathieu .a;/-series,” Integral Transforms Spec.Funct., vol. 16, no. 8, pp. 685–689, 2005. [Online]. Available: http://dx.doi.org/10.1080/10652460500110297[7] T. K. Poga´ny, “Multiple Euler-Maclaurin summation formula,” Mat. Bilten, no. 29, pp. 37–40,2005.[8] T. K. Poga´ny, H. M. Srivastava, and Zˇ. Tomovski, “Some families of Mathieu a-series andalternating Mathieu a-series,” Appl. Math. Comput., vol. 173, no. 1, pp. 69–108, 2006. [Online].Available: http://dx.doi.org/10.1016/j.amc.2005.02.044[9] T. K. Poga´ny and E. Su¨li, “Integral representation for Neumann series of Bessel functions,”Proc. Amer. Math. Soc., vol. 137, no. 7, pp. 2363–2368, 2009. [Online]. Available:http://dx.doi.org/10.1090/S0002-9939-09-09796-2[10] T. K. Poga´ny and Zˇ. Tomovski, “On multiple generalized Mathieu series,” IntegralTransforms Spec. Funct., vol. 17, no. 4, pp. 285–293, 2006. [Online]. Available:http://dx.doi.org/10.1080/10652460500431966[11] F. Qi, “Inequalities for mathieu series,” RGMIA Research Report Collection, vol. 4 (2), Art. 3,pp. 187–193, 2001. [Online]. Available: http://rgmia.vu.edu.au/v4n2.html[12] F. Qi, “An integral expression and some inequalities of Mathieu type series,” Rostock. Math.Kolloq., no. 58, pp. 37–46, 2004.[13] H. M. Srivastava and Zˇ. Tomovski, “Some problems and solutions involving Mathieu’s series andits generalizations,” JIPAM. J. Inequal. Pure Appl. Math., vol. 5, no. 2, pp. Article 45, 13 pp.(electronic), 2004.[14] Zˇ. Tomovski, “New double inequalities for Mathieu type series,” Univ. Beograd. Publ. Elektro-tehn. Fak. Ser. Mat., vol. 15, pp. 80–84, 2004.Author’s addressBiserka Drasˇcˇic´ BanFaculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Studentska 2, CroatiaE-mail address: bdrascic@pfri.hr

On multiple Mathieu ( a,lambda ) –series

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