Some Remarks on Glaisher-Ramanujan Type Integrals
Patkowski Alexander 1, Wolf Marek 2
1 1390 Bumps River Rd. Centerville, MA 02632,
USA E-mail: alexpatk@hotmail.com2 Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences. College of Sciences ul. Wóycickiego 1/3, Auditorium Maximum, (room 113) PL-01-938 Warsaw, Poland
E-mail: m.wolf@uksw.edu.pl
Received:
Received: 06 February 2016; revised: 23 May 2016 ; accepted: 24 May 2016 ; published online: 08 June 2016
DOI: 10.12921/cmst.2016.22.02.005
Abstract:
Some integrals of the Glaisher-Ramanujan type are established in a more general form than in previous studies. As an application we prove some Ramanujan-type series identities, as well as a new formula for the Dirichlet beta function at the value s = 3. We also present a Mathematica program calculating values of beta function at odd positive arguments.
Key words:
Dedekind eta function, Fourier integrals, trigonometric series
References:
[1] M.L. Glasser, Some integrals of the Dedekind η-function, J. Math. Anal. Appl. 354 490-493 (2009).
[2] J.W.L.Glaisher, On the summation by definite integrals of geometric series of the second and higher order, The Quarterly Journal of Pure and Applied Mathematics, 328-343 (1871).
[3] G.H.Hardy,Ramanujan,CambridgeUniversityPress,1940.
[4] The PARI Group, Bordeaux. PARI/GP, version 2.3.2, 2008,
available from http://pari.math.u-bordeaux.fr/.
[5] G. Andrews, R. Askey, and R. Roy. Special Functions, vol- ume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999.
[6] B.C. Berndt, Ramanujans Notebooks, Part II , Springer, 1989.
[7] I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Edited by A.Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition, 2007.
[8] B.C. Berndt and R.J. Evans, Chapter 15 of the Ramanujan Second Notebook : Part II, Modular forms, Acta Arith. 47, 123-142 (1986).
[9] S. Kanemitsu, Y. Tanigawa, M. Toshimoto, On the values of the Riemann zeta-function at rational arguments, Hardy- Ramanujan J. 24 10-18, 2001.
[10] D. Klusch, On Entry 8 of Chapter 15 of Ramanujan’s Note- book II, Acta Arith. 58 59-64, 1991.
[11] D. Shanks and J. Wrench Jr, The Calculation of Certain Dirichlet Series, Math. of Comput. 17, 136-154 (1963).
Some integrals of the Glaisher-Ramanujan type are established in a more general form than in previous studies. As an application we prove some Ramanujan-type series identities, as well as a new formula for the Dirichlet beta function at the value s = 3. We also present a Mathematica program calculating values of beta function at odd positive arguments.
Key words:
Dedekind eta function, Fourier integrals, trigonometric series
References:
[1] M.L. Glasser, Some integrals of the Dedekind η-function, J. Math. Anal. Appl. 354 490-493 (2009).
[2] J.W.L.Glaisher, On the summation by definite integrals of geometric series of the second and higher order, The Quarterly Journal of Pure and Applied Mathematics, 328-343 (1871).
[3] G.H.Hardy,Ramanujan,CambridgeUniversityPress,1940.
[4] The PARI Group, Bordeaux. PARI/GP, version 2.3.2, 2008,
available from http://pari.math.u-bordeaux.fr/.
[5] G. Andrews, R. Askey, and R. Roy. Special Functions, vol- ume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999.
[6] B.C. Berndt, Ramanujans Notebooks, Part II , Springer, 1989.
[7] I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Edited by A.Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition, 2007.
[8] B.C. Berndt and R.J. Evans, Chapter 15 of the Ramanujan Second Notebook : Part II, Modular forms, Acta Arith. 47, 123-142 (1986).
[9] S. Kanemitsu, Y. Tanigawa, M. Toshimoto, On the values of the Riemann zeta-function at rational arguments, Hardy- Ramanujan J. 24 10-18, 2001.
[10] D. Klusch, On Entry 8 of Chapter 15 of Ramanujan’s Note- book II, Acta Arith. 58 59-64, 1991.
[11] D. Shanks and J. Wrench Jr, The Calculation of Certain Dirichlet Series, Math. of Comput. 17, 136-154 (1963).
