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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.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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)

    Article  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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)

    Book  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)

    Article  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)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Anthony Sofo .

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Dedicated to Professor Hari M. Srivastava

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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

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