Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION

  • Cheon, Gi-Sang (Department of Mathematics Sungkyunkwan University) ;
  • El-Mikkawy Moawwad E.A. (Department of mathematics Faculty of Science of Mansoura University)
  • Published : 2007.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.

Keywords

References

  1. V. Adamchik, On Stirling numbers and Euler sums, J. Comput. and Appl. Math. 79 (1997), no. 1, 119-130 https://doi.org/10.1016/S0377-0427(96)00167-7
  2. A. T. Benjamin, D. Gaebler, and R. Gaebler, A combinatorial approach to hyperharmonic numbers, Integers (Elec. J. Combi. Number Theory) 3 (2003), A15, 1-9
  3. S. Chaturvedi and V. Gupta, Identities involving elementary symmetric functions, J. Phys. A 33 (2000), no. 29, L251-L255 https://doi.org/10.1088/0305-4470/33/29/101
  4. W. Chu, Harmonic number identities and Hermite-Pade approximations to the logarithm function, J. Approx. Theory 137 (2005), no. 1, 42-56 https://doi.org/10.1016/j.jat.2005.07.008
  5. W. Chu and L. De Donno, Hypergeometric series and harmonic number identities, Adv. in Appl. Math. 34 (2005), no. 1, 123-137 https://doi.org/10.1016/j.aam.2004.05.003
  6. L. Comtet, Advanced Combinatorics, D. Reidel Pub. Company, 1974
  7. A. Gertsch, Generalized harmonic numbers, C. R. Acad. Sci. Paris, Ser. I Math. 324 (1997), no. 1, 7-10 https://doi.org/10.1016/S0764-4442(97)80094-8
  8. I. M. Gessel, On Miki's identity for Bernoulli numbers, J. Number Theory 110 (2005), no. 1, 75-82 https://doi.org/10.1016/j.jnt.2003.08.010
  9. T. M. Rassias and H. M. Srivastava, Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput. 131 (2002), no. 2-3, 593-605 https://doi.org/10.1016/S0096-3003(01)00172-2
  10. J. M. Santmyer, A Stirling like sequence of rational numbers, Discrete Math. 171 (1997), no. 1-3, 229-235 https://doi.org/10.1016/S0012-365X(96)00082-9
  11. F. Scheid, Numerical Analysis, McGraw-Hill Book Company, 1968
  12. A. J. Sommese, J. Verscheide, and C. W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numerical Analysis 40 (2002), no. 6, 2026-2046 https://doi.org/10.1137/S0036142901397101
  13. http://mathworld.wolfram.com/Newton-GirardFormulas.html

Cited by

  1. Algebraic Comparison of Partial Lists in Bioinformatics vol.7, pp.5, 2012, https://doi.org/10.1371/journal.pone.0036540
  2. Explicit upper bounds for the Stieltjes constants vol.133, pp.3, 2013, https://doi.org/10.1016/j.jnt.2012.09.001
  3. On the harmonic and hyperharmonic Fibonacci numbers vol.2015, pp.1, 2015, https://doi.org/10.1186/s13662-015-0635-z
  4. A family of shifted harmonic sums vol.37, pp.1, 2015, https://doi.org/10.1007/s11139-014-9600-9
  5. An introduction to hyperharmonic numbers vol.46, pp.3, 2015, https://doi.org/10.1080/0020739X.2014.982734
  6. Infinite Series Containing Generalized Harmonic Numbers vol.73, pp.1, 2018, https://doi.org/10.1007/s00025-018-0774-0
  7. Identities between harmonic, hyperharmonic and Daehee numbers vol.2018, pp.1, 2018, https://doi.org/10.1186/s13660-018-1757-0