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Extensions of Some Known Algebraic and Combinatorial Identities

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Abstract

By means of a general partial fraction decomposition expression, we will derive several striking algebraic and combinatorial identities including some results discovered recently by Abel (Aequ Math 94:163–167, 2020). We also recover some famous identities including Chu’s identity and Melzak’s identity.

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References

  1. Abel, U.: New proofs of Melzak’s identity. Aequ. Math. 94(1), 163–167 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adamchik, V.: On stirling numbers and Euler sums. J. Comput. Appl. Math. 79, 119–130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahlgren, S., Ekhad, S.B., Ono, K., Zeilberger, D.: A binomial coefficient identity associated to a conjecture of Beukers. Electron. J. Comb. 5(5), 159–160 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Boyadzhiev, K.N.: Melzak’s formula for arbitrary polynomials. Util. Math. 99, 397–401 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Cheon, G.S., El-Mikkawy, M.E.A.: Generalized harmonic number identities and a related matrix representation. J. Korean Math. Soc. 44, 487–498 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheon, G.S., El-Mikkawy, M.E.A.: Generalized harmonic numbers with Riordan arrays. J. Number Theory. 128, 413–425 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, J.: Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers. J. Inequal. Appl. 49, 1–11 (2013)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Coffey, M.W., Lubbers, N.: On generalized harmonic number sums. Appl. Math. Comput. 217, 689–698 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Feng, C.J., Zhao, F.Z.: Some results for generalized harmonic numbers. Integers. 9, 605–619 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Gould, H.W.: A note on combinatorial identities arising from the Lagrange-Waring interpolation formula. Ars Comb. 86, 281–288 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gould, H.W.: Combinatorial identities. Published by the author, Morgantown, W. VA (1972)

    MATH  Google Scholar 

  18. Guo, D.: Some combinatorial identities concerning harmonic numbers and binomial coefficients. Discret. Math. Lett. 8, 41–48 (2022)

    MathSciNet  MATH  Google Scholar 

  19. Liang, H.: Identities involving generalized harmonic numbers and other special combinatorial sequences. J. Integer Seq. 15, 1–15 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Melzak, Z.A., Gokhale, V.D., Parker, W.V.: Advanced problems and solutions: 4458. Am. Math. Mon. 60(1), 53–54 (1953)

    Article  MathSciNet  Google Scholar 

  21. Mouçouf, M., Zriaa, S.: A new approach for computing the inverse of confluent Vandermonde matrices via Taylor’s expansion. Linear Multilinear Algebra. 7, 5973–5986 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nielsen, N.: Traité élémentaire des Nombres de Bernoulli. Gauthier-Villars, Paris (1923)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Prodinger, H.: Identities involving harmonic numbers that can be of interest to physicists. Util. Math. 83, 291–299 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Quaintance, J., Gould, H.W.: Combinatorial Identities for Stirling Numbers: The Unpublished Notes of HW Gould. World Scientific, Singapore (2015)

    Book  Google Scholar 

  26. Straub, A.: Supercongruences for polynomial analogs of the Apéry numbers. Proc. Am. Math. Soc. 147, 1023–1036 (2019)

    Article  MATH  Google Scholar 

  27. Wang, J., Wei, C.: Derivative operator and summation formulae involving generalized harmonic numbers. J. Math. Anal. Appl. 434, 315–341 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wei, C., Yu, Y., Zhang, H.: Watson-type \(_{3}F_{2}\)-series and summation formulae involving generalized harmonic numbers. J. Differ. Equ. Appl. 24, 1444–1472 (2018)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Zheng, D.Y.: An algebraic identity on q-Apéry numbers. Discrete Math. 311, 23–24 (2011)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are sincerely grateful to the anonymous referee for his/her careful reading, many critical comments and valuable suggestions that contribute significantly to improving the manuscript during the revision.

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  1. Department of Mathematics, Faculty of Science, Chouaib Doukkali University, Eljadida, Morocco

    Said Zriaa & Mohammed Mouçouf

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  1. Said Zriaa
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  2. Mohammed Mouçouf
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Zriaa, S., Mouçouf, M. Extensions of Some Known Algebraic and Combinatorial Identities. Results Math 78, 176 (2023). https://doi.org/10.1007/s00025-023-01944-0

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