Skip to main content
SpringerLink
Log in

Integrals involving the Legendre Chi function

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper we investigate the representation of integrals involving the Legendre Chi function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alzer, H., Choi, J.: Four parametric linear Euler sums. J. Math. Anal. Appl 484(1), 123661 (2020)

    Article  MathSciNet  Google Scholar 

  2. Batir, N.: On some combinatorial identities and harmonic sums. Int. J. Number Theory 13(7), 1695–1709 (2017)

    Article  MathSciNet  Google Scholar 

  3. Boersma, J., Dempsey, J.P.: On the numerical evaluation of Legendre’s chi-function. Math. Comput. 59(199), 157–163 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Borwein, D., Borwein, J.M., Bradley, D.M.: Parametric Euler sum identities. J. Math. Anal. Appl. 316(1), 328–338 (2006)

    Article  MathSciNet  Google Scholar 

  5. Borwein, D., Borwein, J.M., Girgensohn, R.: Explicit evaluation of Euler sums. Proc. Edinb. Math. Soc. (2) 38(2), 277–294 (1995)

    Article  MathSciNet  Google Scholar 

  6. Choi, J.: Some identities involving the Legendre’s chi-function. Commun. Korean Math. Soc. 22(2), 219–225 (2007)

    Article  MathSciNet  Google Scholar 

  7. Cvijović, D., Klinowski, J.: Values of the Legendre chi and Hurwitz zeta functions at rational arguments. Math. Comput. 68(228), 1623–1630 (1999)

    Article  MathSciNet  Google Scholar 

  8. Cvijović, D.: Integral representations of the Legendre chi function. J. Math. Anal. Appl. 332(2), 1056–1062 (2007)

    Article  MathSciNet  Google Scholar 

  9. Cvijović, D.: Exponential and trigonometric sums associated with the Lerch zeta and Legendre chi functions. Comput. Math. Appl. 59(4), 1484–1490 (2010)

    Article  MathSciNet  Google Scholar 

  10. Devoto, A., Duke, D.W.: Table of integrals and formulae for Feynman diagram calculations. Riv. Nuovo Cimento (3) 7(6), 1–39 (1984)

    Article  MathSciNet  Google Scholar 

  11. Flajolet, P., Salvy, B.: Euler sums and contour integral representations. Exp. Math. 7(1), 15–35 (1998)

    Article  MathSciNet  Google Scholar 

  12. Freitas, P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74(251), 1425–1440 (2005)

    Article  MathSciNet  Google Scholar 

  13. Georghiou, C., Philippou, A.N.: Harmonic sums and the zeta function. Fibo. Q. 21, 29–36 (1983)

    MathSciNet  MATH  Google Scholar 

  14. Lewin, R.: Polylogarithms and Associated Functions. North Holland, New York (1981)

    MATH  Google Scholar 

  15. Nimbran, A.S., Sofo, A.: New interesting Euler sums. J. Class. Anal. 15(1), 9–22 (2019)

    Article  MathSciNet  Google Scholar 

  16. Sofo, A.: Integrals of polylogarithmic functions with alternating argument. Asian-Eur. J. Math. 13(7), 2050125 (2020)

    Article  MathSciNet  Google Scholar 

  17. Sofo, A.: Integral identities for sums. Math. Commun. 13(2), 303–309 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Sofo, A., Srivastava, H.M.: A family of shifted harmonic sums. Ramanujan J. 37(1), 89–108 (2015)

    Article  MathSciNet  Google Scholar 

  19. Sofo, A.: New classes of harmonic number identities. J. Integer Seq. 15(7), Article 12.7.4 (2012)

  20. Sofo, A., Cvijović, D.: Extensions of Euler harmonic sums. Appl. Anal. Discrete Math. 6(2), 317–328 (2012)

    Article  MathSciNet  Google Scholar 

  21. Sofo, A.: Shifted harmonic sums of order two. Commun. Korean Math. Soc. 29(2), 239–255 (2014)

    Article  MathSciNet  Google Scholar 

  22. Sofo, A.: General order Euler sums with rational argument. Integr. Transforms Spec. Funct. 30(12), 978–991 (2019)

    Article  MathSciNet  Google Scholar 

  23. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer, Dordrecht, x+388 pp. ISBN:0-7923-7054-6 (2001)

  24. Vălean, C.I.: (Almost) impossible integrals, sums, and series. Problem Books in Mathematics. Springer, Cham, xxxviii+539 pp. ISBN:978-3-030-02461-1; 978-3-030-02462-8 41-01 (00A07 26-01 33F05) (2019)

Download references

Author information

Authors and Affiliations

  1. College of Engineering and Science, Victoria University, Melbourne, Australia

    A. Sofo

Authors
  1. A. Sofo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Sofo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sofo, A. Integrals involving the Legendre Chi function. RACSAM 115, 24 (2021). https://doi.org/10.1007/s13398-020-00963-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-020-00963-8

Keywords

Mathematics Subject Classification

Search

Navigation