Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal \(3\times 3\)-matrices

Volume 10, Issue 2, pp 560--574 http://dx.doi.org/10.22436/jnsa.010.02.20

Publication Date: February 20, 2017 Submission Date: September 02, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2189 Downloads
• 3456 Views

Authors

I. A. Shilin - Department of Mathematics, Sholokhov Moscow State University for the Humanities, Verhnya Radishevskaya 16-18, Moscow 109240, Russia. - Department of Energetics, University of Economics and Energetics, Kirovogradskaya ul. 11-1, Moscow 117587, Russia. J. Choi - Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea.

Abstract

The authors derive the matrix elements of the linear operators which appear under the representation of the group SO(2, 1) and correspond to some diagonal or block-diagonal matrices belonging to the above group. Then, by applying these matrix elements, that is, from a group theoretical point of view, the authors show how certain interesting integral and series representations of the Whittaker function of the second kind and some formulas for the (basic and modified) Bessel functions can be obtained. A special case of one of the results presented here is indicated to be also a special one of a known formula.

Share and Cite

Keywords

• Whittaker function
• Bessel functions
• Macdonald function
• semisimple Lie group SO(2، 1)
• matrix elements of representation.
• Whittaker function
• Bessel functions
• Macdonald function
• semisimple Lie group SO(2، 1)
• matrix elements of representation.

MSC

•  33C10
•  33C80
•  33B15
•  33C05

References

• [1] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger., Academic Press, Amsterdam (2007)
  • MathSciNet
  • Google Scholar


• [2] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev , Integrals and Series, Vol. 1: Elementary Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, New York (1986)
  • MATH


• [3] I. A. Shilin, Double SO(2, 1)-integrals and formulas for Whittaker functions, Russian Math., 56 (2012), 47–56.
  • View Article
  • Google Scholar
  • MATH


• [4] I. A. Shilin, J. S. Choi, Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions, Integral Transforms Spec. Funct., 25 (2014), 374–383.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] I. A. Shilin, J. Choi , Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald function, Abstr. Appl. Anal., 2014 (2014), 8 page.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] I. A. Shilin, J. Choi, Transformations of bases related to the six-dimensional split orthogonal group and special functions, , (submitted),


• [7] H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers , Amsterdam (2012)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] N. J. Vilenkin, M. A. Sleinikova, Integral relations for the Whittakers functions and the representations of the threedimensional Lorentz group, Math. USSR Sb., 10 (1970), 173–180.
  • View Article
  • Google Scholar