Skip to main content

Identities for Reciprocal Binomials

Abstract

Euler’s results related to the sum of the ratios of harmonic numbers and binomial coefficients are investigated in this paper. We give a particular example involving quartic binomial coefficients.

Keywords

  • Harmonic Number
  • Binomial Coefficients
  • Euler Sums
  • Binomial Sums
  • Elegant Formula

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4939-0258-3_8
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-1-4939-0258-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   199.99
Price excludes VAT (USA)

References

  1. Alzer, H., Karaannakis D., Srivastava, H.M.: Series representations of some mathematical constants. J. Math. Anal. Appl. 320, 145–162 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Basu, A.: A new method in the study of Euler sums. Ramanujan J. 16, 7–24 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Brychkov, Yu.A.: Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Chapman and Hall/CRC Press, Boca Raton (2008)

    Google Scholar 

  4. Cho, Y., Jung, M., Choi, J., Srivastava, H.M.: Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function. Int. J. Comput. Math. 83, 461–472 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Choi, J.: Certain summation formulas involving harmonic numbers and generalized harmonic numbers. Appl. Math. Comput. 218, 734–740 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Choi, J., Srivastava, H.M.: Sums associated with the zeta function. J. Math. Anal. Appl. 206, 103–120 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Choi, J., Srivastava, H.M.: Explicit evaluation of Euler and related sums. Ramanujan J. 10, 51–70 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Choi, J., Srivastava, H.M.: Some summation formulas involving harmonic numbers and generalized harmonic numbers. Math. Comp. Modelling. 54, 2220–2234 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Chu, W.: Summation formulae involving harmonic numbers. Filomat 26, 143–152 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Dil, A., Kurt, V.: Polynomials related to harmonic numbers and evaluation of harmonic number series I. Integers 12, A38 (2012)

    MathSciNet  Google Scholar 

  11. Lin, S., Tu, S., Hsieh, T., Srivastava, H.M.: Some finite and infinite sums associated with the digamma and related functions. J. Fract. Calc. 22, 103–114 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Lin, S., Hsieh, T., Srivastava, H.M.: Some families of multiple infinite sums associated with the digamma and related functions. J. Fract. Calc. 24, 77–85 (2003)

    MATH  MathSciNet  Google Scholar 

  13. Liu, H., Wang, W.: Harmonic number identities via hypergeometric series and Bell polynomials. Integral Transforms Spec. Funct. 23(1), 49–68 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Munarini, E.: Riordan matrices and sums of harmonic numbers. Appl. Anal. Discrete Math. 5, 176–200 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Petojević, A., Srivastava, H.M.: Computation of Euler’s type sums of the products of Bernoulli numbers. Appl. Math. Lett. 22, 796–801 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Rassias, T.M., Srivastava, H.M.: Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers. Appl. Math. Comput. 131, 593–605 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Sofo, A.: Computational Techniques for the Summation of Series. Kluwer Academic/Plenum Publishers, New York (2003)

    CrossRef  MATH  Google Scholar 

  18. Sofo, A.: Integral forms of sums associated with Harmonic numbers. Appl. Math. Comput. 207, 365–372 (2009)

    CrossRef  MATH  Google Scholar 

  19. Sofo, A.: Harmonic numbers and double binomial coefficients. Integral Transforms Spec. Funct. 20(11), 847–857 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Sofo, A.: Sums of derivatives of binomial coefficients. Adv. Appl. Math. 42, 123–134 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. Sofo, A.: Harmonic sums and integral representations. J. Appl. Anal. 16, 265–277 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Sofo, A.: Summation formula involving harmonic numbers. Anal. Math. 37(1), 51–64 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Sofo, A., Srivastava, H.M.: Identities for the harmonic numbers and binomial coefficients. Ramanujan J. 25, 93–113 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. Sondow, J., Weisstein, E.W.: Harmonic number. From MathWorld-A Wolfram Web Resources. http://mathworld.wolfram.com/HarmonicNumber.html

  26. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, London (2001)

    CrossRef  MATH  Google Scholar 

  27. Wei, C., Gong, D., Wang, Q.: Chu-Vandermonde convolution and harmonic number identities. Integral Transforms and Spec. Funct. (2012). doi: 10.1080/10652469.2012.689762

    Google Scholar 

  28. Wu, T., Leu, S., Tu, S., Srivastava, H.M.: A certain class of infinite sums associated with digamma functions. Appl. Math. Comput. 105, 1–9 (1999)

    CrossRef  MathSciNet  Google Scholar 

  29. Wu, T., Tu, S., Srivastava, H.M.: Some combinatorial series identities associated with the digamma function and harmonic numbers. Appl. Math. Lett. 13, 101–106 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anthony Sofo .

Editor information

Editors and Affiliations

Additional information

Dedicated to Professor Hari M. Srivastava

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Sofo, A. (2014). Identities for Reciprocal Binomials. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_8

Download citation