Research Article
BibTex RIS Cite

On generalized Mathieu series and its companions

Year 2019, Volume: 48 Issue: 5, 1367 - 1387, 08.10.2019

Abstract

Integral representations for a generalized Mathieu series and its companions are used to obtain bounds for their corresponding series. The bounds are procured mainly using results pertaining to the Čebyšev functional. The relationship to Zeta type functions are also examined. It is demonstrated that the Zeta companion relations are a particular case of the generalised Mathieu companions.

References

  • [1] H. Alzer, J.L. Brenner and O.G. Ruehr, On Mathieu’s inequality, J. Math. Anal. Appl. 218, 607-610, 1998.
  • [2] A. Bagdasaryan, A new lower bound for Mathieu’s series, New Zealand J. Math. 44, 75-81, 2014.
  • [3] Á. Baricz, P.L. Butzer, T.K. Pogány, Alternating Mathieu series, Hilbert-Eisenstein series and their generalized Omega functions, Analytic number theory, approximation theory, and special functions, 775-808, Springer, New York, 2014
  • [4] P.S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998.
  • [5] P. Cerone, On an identity of the Chebychev functional and some ramifications, J. Inequal. Pure & Appl. Math. 3 (1), 2002.
  • [6] P. Cerone, Special functions approximation and bounds via integral representation, Advances in Inequalities for Special Functions, Nova Science Publishers Inc., 1-36, 2008.
  • [7] P. Cerone and S.S. Dragomir, New upper and lower bounds for the Chebyshev func- tional, J. Inequal. Pure & Appl. Math. 3(5), 2002.
  • [8] P. Cerone and C. Lenard, On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5), 11 pages, 2003.
  • [9] N. Elezović, H.M. Srivastava and Ž. Tomovski, Integral representations and integral transforms of some families of Mathieu type series, Integral Transforms Spec. Funct. 19 (7), 481-495, 2008.
  • [10] O.E. Emersleben, Über die Reihe $\sum_{k=1}^{\infty}k(k^{2}+c^{2})^{-2}$, Math. Ann., 125, 165-171, 1952.
  • [11] R. Frontczak, Some remarks on the Mathieu series, ISRN Appl. Math. 8 pages., Art. ID 985782, 2014.
  • [12] I. Gavrea, Some remarks on Mathieu’s series, Mathematical Analysis and Approxi- mation Theory, 113-117, Burg Verlag, 2002.
  • [13] I.S. Gradshtein and I.M. Ryzhik, Table in Integrals, Series and Products, Academic Press, New York, 1980.
  • [14] G. Grüss, Über das maximum des absoluten Betrages von $ \frac{1}{b-a}\int_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{b-a} \int_{a}^{b}f\left( x\right) dx\cdot \frac{1}{b-a}\int_{a}^{b}g\left(x\right) dx$, Math. Z., 39, 215-226, 1934.
  • [15] B.-N. Guo, Note on Mathieu’s inequality, RGMIA Res. Rep. Coll. 3 (3), Article 5, 2000.
  • [16] L. Landau, Monotonicity and bounds on Bessel functions, Electron. J. Differ. Eq., Conf. 04, 147-154, 2002.
  • [17] Q.-M. Luo, B.-N. Guo and F. Qi, On evaluation of Riemann zeta function $\zeta \left( s\right)$, RGMIA Res. Rep. Coll. 6(1), Article 8, 2003.
  • [18] E. Mathieu, Traité de physique mathématique, VI–VII: Théorie de l’élasticité des corps solides, Gauthier-Villars, Paris, 1890.
  • [19] V.V. Meleshko, Equilibrium of elastic rectangle: Mathieu-Inglis-Pickett solution re- visited, J. Elasticity 40 (3), 207-238, 1995.
  • [20] V.V. Meleshko, Biharmonic problem in a rectangle, Appl. Sci. Res. 58, 217-249, 1997.
  • [21] V.V. Meleshko, Bending of an elastic rectangular clamped plate: exact versus "engi- neering” solutions, J. Elasticity 48 (1), 1-50, 1997.
  • [22] G.V. Milovanović and T. K. Pogány, New integral forms of generalized Mathieu series and related applications, Appl. Anal. Discrete Math. 7 (1), 180-192, 2013.
  • [23] D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Anal- ysis, Kluwer Academic Publishers, Dordrecht, 1993.
  • [24] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Math- ematical Functions, National Bureau of Standards and Technology, Cambridge Uni- versity Press, New York, 2010
  • [25] T.K. Pogány, H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a- seriesand alternating Mathieu a-series, Appl. Math. Comput. 173, 69-108, 2006.
  • [26] T.K. Pogány and Ž. Tomovski, Bounds improvement for alternating Mathieu type series, J. Math. Inequal. 4 (3), 315-324, 2010.
  • [27] F. Qi, Inequalities for Mathieu’s series, RGMIA Res. Rep. Coll. 4 (2), 1-7 Article 3, 2001.
  • [28] F. Qi, Integral expressions and inequalities of Mathieu type series, RGMIA Res. Rep. Coll. 6 (2), Article 5, 2003.
  • [29] F. Qi, Ch.-P. Chen and B.-N. Guo, Notes on double inequalities of Mathieu’s series, Int. J. Math. Math. Sci. 2005, 2547-2554, 2005.
  • [30] M. Srivastava, Ž. Tomovski and D. Leškovski, Some families of Mathieu type se- ries and Hurwitz-Lerch zeta functions and associated probability distributions, Appl. Comput. Math. 14 (3), 349-380, 2015.
  • [31] Ž. Tomovski and K. Trenčevski, On an open problem of Bai-Ni Guo and Feng Qi, J. Inequal. Pure Appl. Math. 4 (2), 2003.
  • [32] G.N. Watson, A treatise on the theory of Bessel functions 2nd Edn., Cambridge University Press, 1966.
  • [33] J.E. Wilkins, Jr., Solution of Problem 97-1, Siam Rev. , 1998.
Year 2019, Volume: 48 Issue: 5, 1367 - 1387, 08.10.2019

Abstract

References

  • [1] H. Alzer, J.L. Brenner and O.G. Ruehr, On Mathieu’s inequality, J. Math. Anal. Appl. 218, 607-610, 1998.
  • [2] A. Bagdasaryan, A new lower bound for Mathieu’s series, New Zealand J. Math. 44, 75-81, 2014.
  • [3] Á. Baricz, P.L. Butzer, T.K. Pogány, Alternating Mathieu series, Hilbert-Eisenstein series and their generalized Omega functions, Analytic number theory, approximation theory, and special functions, 775-808, Springer, New York, 2014
  • [4] P.S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998.
  • [5] P. Cerone, On an identity of the Chebychev functional and some ramifications, J. Inequal. Pure & Appl. Math. 3 (1), 2002.
  • [6] P. Cerone, Special functions approximation and bounds via integral representation, Advances in Inequalities for Special Functions, Nova Science Publishers Inc., 1-36, 2008.
  • [7] P. Cerone and S.S. Dragomir, New upper and lower bounds for the Chebyshev func- tional, J. Inequal. Pure & Appl. Math. 3(5), 2002.
  • [8] P. Cerone and C. Lenard, On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5), 11 pages, 2003.
  • [9] N. Elezović, H.M. Srivastava and Ž. Tomovski, Integral representations and integral transforms of some families of Mathieu type series, Integral Transforms Spec. Funct. 19 (7), 481-495, 2008.
  • [10] O.E. Emersleben, Über die Reihe $\sum_{k=1}^{\infty}k(k^{2}+c^{2})^{-2}$, Math. Ann., 125, 165-171, 1952.
  • [11] R. Frontczak, Some remarks on the Mathieu series, ISRN Appl. Math. 8 pages., Art. ID 985782, 2014.
  • [12] I. Gavrea, Some remarks on Mathieu’s series, Mathematical Analysis and Approxi- mation Theory, 113-117, Burg Verlag, 2002.
  • [13] I.S. Gradshtein and I.M. Ryzhik, Table in Integrals, Series and Products, Academic Press, New York, 1980.
  • [14] G. Grüss, Über das maximum des absoluten Betrages von $ \frac{1}{b-a}\int_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{b-a} \int_{a}^{b}f\left( x\right) dx\cdot \frac{1}{b-a}\int_{a}^{b}g\left(x\right) dx$, Math. Z., 39, 215-226, 1934.
  • [15] B.-N. Guo, Note on Mathieu’s inequality, RGMIA Res. Rep. Coll. 3 (3), Article 5, 2000.
  • [16] L. Landau, Monotonicity and bounds on Bessel functions, Electron. J. Differ. Eq., Conf. 04, 147-154, 2002.
  • [17] Q.-M. Luo, B.-N. Guo and F. Qi, On evaluation of Riemann zeta function $\zeta \left( s\right)$, RGMIA Res. Rep. Coll. 6(1), Article 8, 2003.
  • [18] E. Mathieu, Traité de physique mathématique, VI–VII: Théorie de l’élasticité des corps solides, Gauthier-Villars, Paris, 1890.
  • [19] V.V. Meleshko, Equilibrium of elastic rectangle: Mathieu-Inglis-Pickett solution re- visited, J. Elasticity 40 (3), 207-238, 1995.
  • [20] V.V. Meleshko, Biharmonic problem in a rectangle, Appl. Sci. Res. 58, 217-249, 1997.
  • [21] V.V. Meleshko, Bending of an elastic rectangular clamped plate: exact versus "engi- neering” solutions, J. Elasticity 48 (1), 1-50, 1997.
  • [22] G.V. Milovanović and T. K. Pogány, New integral forms of generalized Mathieu series and related applications, Appl. Anal. Discrete Math. 7 (1), 180-192, 2013.
  • [23] D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Anal- ysis, Kluwer Academic Publishers, Dordrecht, 1993.
  • [24] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Math- ematical Functions, National Bureau of Standards and Technology, Cambridge Uni- versity Press, New York, 2010
  • [25] T.K. Pogány, H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a- seriesand alternating Mathieu a-series, Appl. Math. Comput. 173, 69-108, 2006.
  • [26] T.K. Pogány and Ž. Tomovski, Bounds improvement for alternating Mathieu type series, J. Math. Inequal. 4 (3), 315-324, 2010.
  • [27] F. Qi, Inequalities for Mathieu’s series, RGMIA Res. Rep. Coll. 4 (2), 1-7 Article 3, 2001.
  • [28] F. Qi, Integral expressions and inequalities of Mathieu type series, RGMIA Res. Rep. Coll. 6 (2), Article 5, 2003.
  • [29] F. Qi, Ch.-P. Chen and B.-N. Guo, Notes on double inequalities of Mathieu’s series, Int. J. Math. Math. Sci. 2005, 2547-2554, 2005.
  • [30] M. Srivastava, Ž. Tomovski and D. Leškovski, Some families of Mathieu type se- ries and Hurwitz-Lerch zeta functions and associated probability distributions, Appl. Comput. Math. 14 (3), 349-380, 2015.
  • [31] Ž. Tomovski and K. Trenčevski, On an open problem of Bai-Ni Guo and Feng Qi, J. Inequal. Pure Appl. Math. 4 (2), 2003.
  • [32] G.N. Watson, A treatise on the theory of Bessel functions 2nd Edn., Cambridge University Press, 1966.
  • [33] J.E. Wilkins, Jr., Solution of Problem 97-1, Siam Rev. , 1998.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

P. Cerone 0000-0002-0271-431X

C. Lenard This is me 0000-0003-4819-0125

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Cerone, P., & Lenard, C. (2019). On generalized Mathieu series and its companions. Hacettepe Journal of Mathematics and Statistics, 48(5), 1367-1387.
AMA Cerone P, Lenard C. On generalized Mathieu series and its companions. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1367-1387.
Chicago Cerone, P., and C. Lenard. “On Generalized Mathieu Series and Its Companions”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1367-87.
EndNote Cerone P, Lenard C (October 1, 2019) On generalized Mathieu series and its companions. Hacettepe Journal of Mathematics and Statistics 48 5 1367–1387.
IEEE P. Cerone and C. Lenard, “On generalized Mathieu series and its companions”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1367–1387, 2019.
ISNAD Cerone, P. - Lenard, C. “On Generalized Mathieu Series and Its Companions”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1367-1387.
JAMA Cerone P, Lenard C. On generalized Mathieu series and its companions. Hacettepe Journal of Mathematics and Statistics. 2019;48:1367–1387.
MLA Cerone, P. and C. Lenard. “On Generalized Mathieu Series and Its Companions”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1367-8.
Vancouver Cerone P, Lenard C. On generalized Mathieu series and its companions. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1367-8.