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Harmonic number identities via the Newton–Andrews method

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Abstract

By means of the Bell polynomials, we establish explicit expressions of the higher-order derivatives of the binomial coefficient \(\binom{x+n}{m}\) and its reciprocal \(\binom{x+n}{m}^{-1}\), and extend the application field of the Newton–Andrews method. As examples, we apply the results to the Chu–Vandermonde–Gauss formula and the Dougall–Dixon theorem and obtain a series of harmonic number identities. This paper generalizes some works presented before and provides a way to establish infinite harmonic number identities.

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References

  1. Andrews, G.E., Uchimura, K.: Identities in combinatorics. IV. Differentiation and harmonic numbers. Util. Math. 28, 265–269 (1985)

    MathSciNet  MATH  Google Scholar 

  2. Bailey, W.N.: Generalized Hypergeometric Series. Stechert-Hafner, New York (1964)

    Google Scholar 

  3. Chen, X., Chu, W.: The Gauss 2 F 1(1)-summation theorem and harmonic number identities. Integral Transforms Spec. Funct. 20(11–12), 925–935 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X., Chu, W.: Dixon’s 3 F 2(1)-series and identities involving harmonic numbers and the Riemann zeta function. Discrete Math. 310(1), 83–91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chu, W.: Hypergeometric series and the Riemann zeta function. Acta Arith. 82(2), 103–118 (1997)

    MathSciNet  Google Scholar 

  6. Chu, W.: A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers. Electron. J. Comb. 11(1), N15 (2004). 3 pp

    Google Scholar 

  7. Chu, W.: Harmonic number identities and Hermite–Padé approximations to the logarithm function. J. Approx. Theory 137(1), 42–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chu, W.: Hypergeometric approach to Weideman’s conjecture. Arch. Math. 87(5), 400–406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chu, W.: Partial-fraction decompositions and harmonic number identities. J. Comb. Math. Comb. Comput. 60, 139–153 (2007)

    MATH  Google Scholar 

  10. Chu, W.: Infinite series identities on harmonic numbers. Results Math. 61(3–4), 209–221 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chu, W.: Summation formulae involving harmonic numbers. Filomat 26(1), 143–152 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chu, W., De Donno, L.: Hypergeometric series and harmonic number identities. Adv. Appl. Math. 34(1), 123–137 (2005)

    Article  MATH  Google Scholar 

  13. Chu, W., Fu, A.M.: Dougall–Dixon formula and harmonic number identities. Ramanujan J. 18(1), 11–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chu, W., Yan, Q.: Combinatorial identities on binomial coefficients and harmonic numbers. Util. Math. 75, 51–66 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Chu, W., Zheng, D.-Y.: Infinite series with harmonic numbers and central binomial coefficients. Int. J. Number Theory 5(3), 429–448 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974)

    Book  MATH  Google Scholar 

  17. Díaz-Barrero, J.L.: Problem 11164: a recurrent identity. Am. Math. Mon. 112, 568 (2005)

    Article  Google Scholar 

  18. Díaz-Barrero, J.L., Gibergans-Báguena, J., Popescu, P.G.: Some identities involving rational sums. Appl. Anal. Discrete Math. 1(2), 397–402 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Driver, K., Prodinger, H., Schneider, C., Weideman, J.A.C.: Padé approximations to the logarithm. II. Identities, recurrences, and symbolic computation. Ramanujan J. 11(2), 139–158 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Driver, K., Prodinger, H., Schneider, C., Weideman, J.A.C.: Padé approximations to the logarithm. III. Alternative methods and additional results. Ramanujan J. 12(3), 299–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gould, H.W.: Higher order extensions of Melzak’s formula. Util. Math. 72, 23–32 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Kirschenhofer, P.: A note on alternating sums. Electron. J. Comb. 3(2) (1996). Research Paper 7. Approx. 10 pp

  23. Kirschenhofer, P., Larcombe, P.J.: On a class of recursive-based binomial coefficient identities involving harmonic numbers. Util. Math. 73, 105–115 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Larcombe, P.J., Fennessey, E.J., Koepf, W.A.: Integral proofs of two alternating sign binomial coefficient identities. Util. Math. 66, 93–103 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Larcombe, P.J., Larsen, M.E., Fennessey, E.J.: On two classes of identities involving harmonic numbers. Util. Math. 67, 65–80 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Larsen, M.E., Larcombe, P.J.: Some binomial coefficient identities of specific and general type. Util. Math. 74, 33–53 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Osburn, R., Schneider, C.: Gaussian hypergeometric series and supercongruences. Math. Comput. 78(265), 275–292 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Paule, P., Schneider, C.: Computer proofs of a new family of harmonic number identities. Adv. Appl. Math. 31(2), 359–378 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Prodinger, H.: Human proofs of identities by Osburn and Schneider. Integers 8, A10 (2008). 8 pp

    MathSciNet  Google Scholar 

  31. Riordan, J.: An Introduction to Combinatorial Analysis, Reprint of the 1958 Original. Dover, Mineola (2002)

    Google Scholar 

  32. Sofo, A.: Some more identities involving rational sums. Appl. Anal. Discrete Math. 2(1), 56–64 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sofo, A.: Harmonic numbers and double binomial coefficients. Integral Transforms Spec. Funct. 20(11–12), 847–857 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sofo, A.: Integral forms of sums associated with harmonic numbers. Appl. Math. Comput. 207(2), 365–372 (2009)

    Article  MATH  Google Scholar 

  35. Sofo, A.: Sums of derivatives of binomial coefficients. Adv. Appl. Math. 42(1), 123–134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sofo, A.: Harmonic sums and integral representations. J. Appl. Anal. 16(2), 265–277 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Sofo, A., Srivastava, H.M.: Identities for the harmonic numbers and binomial coefficients. Ramanujan J. 25(1), 93–113 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Spieß, J.: Some identities involving harmonic numbers. Math. Comput. 55(192), 839–863 (1990)

    Article  MATH  Google Scholar 

  40. Srivastava, R.: Some combinatorial series identities and rational sums. Integral Transforms Spec. Funct. 20(1–2), 83–91 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Srivastava, R.: Some families of combinatorial and other series identities and their applications. Appl. Math. Comput. 218(3), 1077–1083 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, W.: Riordan arrays and harmonic number identities. Comput. Math. Appl. 60(5), 1494–1509 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wei, C., Gu, Q.: q-Generalizations of a family of harmonic number identities. Adv. Appl. Math. 45(1), 24–27 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Weideman, J.A.C.: Padé approximations to the logarithm. I. Derivation via differential equations. Quaest. Math. 28(3), 375–390 (2005)

    Article  MathSciNet  Google Scholar 

  45. Zave, D.A.: A series expansion involving the harmonic numbers. Inf. Process. Lett. 5(3), 75–77 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 80(2), 207–211 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zheng, D.-Y.: Further summation formulae related to generalized harmonic numbers. J. Math. Anal. Appl. 335(1), 692–706 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, P.R. China

    Weiping Wang

  2. Department of Mathematics, Dalian Maritime University, Dalian, 116026, P.R. China

    Cangzhi Jia

Authors
  1. Weiping Wang
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  2. Cangzhi Jia
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Correspondence to Weiping Wang.

Additional information

The first author is supported by the National Natural Science Foundation of China under Grant 11001243, the Zhejiang Provincial Natural Science Foundation of China under Grant LY13A010016, the “521” Talents Program of Zhejiang Sci-Tech University (ZSTU), and the Science Foundation of Zhejiang Sci-Tech University under Grant 1013817-Y.

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Wang, W., Jia, C. Harmonic number identities via the Newton–Andrews method. Ramanujan J 35, 263–285 (2014). https://doi.org/10.1007/s11139-013-9511-1

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  • DOI: https://doi.org/10.1007/s11139-013-9511-1

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