Some Remarks on Glaisher-Ramanujan Type Integrals

• Autorzy: Patkowski A. E. , Wolf M.

Identyfikatory

DOI

10.12921/cmst.2016.22.02.005

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Fourier integrals; Dedekind eta function; trigonometric series

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2016
• Tom: Vol. 22, No. 2
• Strony: 103--108
• Opis fizyczny: Bibliogr. 11 poz., rys.

Twórcy

autor

Patkowski A. E.

• 390 Bumps River Rd. Centerville, MA 02632, USA

autor

Wolf M.

• 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

Bibliografia

• [1] M.L. Glasser, Some integrals of the Dedekind _-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, Cambridge University Press, 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, volume 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 Notebook 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).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.