대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제29권2호
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  • Pages.269-283
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  • 2014
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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FURTHER EXPANSION AND SUMMATION FORMULAS INVOLVING THE HYPERHARMONIC FUNCTION

  • Gaboury, Sebastien (Department of Mathematics and Computer Science University of Quebec at Chicoutimi)
  • 투고 : 2013.12.17
  • 발행 : 2014.04.30
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초록

The aim of the paper is to present several new relationships involving the hyperharmonic function introduced by Mez$\ddot{o}$ in (I. Mez$\ddot{o}$, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4, 2009) which is an analytic extension of the hyperharmonic numbers. These relations are obtained by using some fractional calculus theorems as Leibniz rules and Taylor like series expansions.

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참고문헌

  1. A. T. Benjamin and D. Gaebler, and R. Gaebler, A combinatorial approach to hyperharmonic numbers, Integers 3 (2003), 1-9.
  2. J. H. Conway and R. K. Guy, The Book of Numbers, Sringer-Verlag, New York, 1996.
  3. A. Erdelyi, An integral equation involving Legendre functions, J. Soc. Indust. Appl. Math. 12 (1964), 15-30. https://doi.org/10.1137/0112002
  4. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions. Vols. 1-3, McGraw-Hill, New York, 1953.
  5. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms. Vols. 1-2, McGraw-Hill, New York, 1953.
  6. G. H. Hardy, Riemann's forms of Taylor's series, J. London. Math. Soc. 20 (1945), 48-57.
  7. O. Heaviside, Electromagnetic Theory. Vol. 2, Dover, New York, 1950.
  8. J.-L. Lavoie, T. J. Osler, and R. Tremblay, Fundamental properties of fractional derivatives via Pochhammer integrals, Lecture Notes in Mathematics, Springer-Verlag, 1976.
  9. J. Liouville, Memoire sur le calcul des differentielles a indices quelconques, J. de l'Ecole Polytechnique 13 (1832), 71-162.
  10. I. Mezo, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4 (2009), 9 pp.
  11. I. Mezo and A. Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), no. 2, 360-369. https://doi.org/10.1016/j.jnt.2009.08.005
  12. K. S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, New York, Chichester, Brisbane, Toronto and Singapore, John Wiley and Sons, Wiley, New York, 1993.
  13. T. J. Osler, Fractional derivatives of a composite function, SIAM J. Math. Anal. 1 (1970), 288-293. https://doi.org/10.1137/0501026
  14. T. J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math. 18 (1970), 658-674. https://doi.org/10.1137/0118059
  15. T. J. Osler, Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives, PhD Thesis, New York University, 1970.
  16. T. J. Osler, Mathematical notes: Fractional derivatives Leibniz rule, Amer. Math. Monthly 78 (1971), no. 6, 645-649. https://doi.org/10.2307/2316573
  17. T. J. Osler, Taylor's series generalized for fractional derivatives and applications, SIAM J. Math. Anal. 2 (1971), 37-48. https://doi.org/10.1137/0502004
  18. T. J. Osler, An integral analogue of Taylor's series and its use in computing Fourier's transform, Math. Comput. 26 (1972), 449-460.
  19. B. Riemann, Versuch einer allgemeinen Auffasung der Integration und Differentiation, The Collection Works of Bernhard Riemann, Dover New York, 1953.
  20. M. Riesz, L'integrale de Riemann-Liouville et le probleme de Cauchy, Acta Math. 81 (1949), 1-233. https://doi.org/10.1007/BF02395016
  21. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press London, 1966.
  22. R. Tremblay, Une contribution a la theorie de la derivee fractionnaire, PhD Thesis, Laval University, Canada, 1974.
  23. J.-L. Lavoie, T. J. Osler, and R. Tremblay, Fundamental properties of fractional derivatives via Pochhammer integrals, Lecture Notes in Mathematics, Springer-Verlag, 1976.
  24. R. Tremblay and B. J. Fugere, The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions, Appl. Math. Comput. 187 (2007), no. 1, 507-529. https://doi.org/10.1016/j.amc.2006.09.076
  25. R. Tremblay, S. Gaboury, and B.-J. Fugere, A new Leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), no. 2, 111-128. https://doi.org/10.1080/10652469.2012.668904
  26. R. Tremblay, S. Gaboury, and B.-J. Fugere, Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives, Integral Transforms Spec. Funct. 24 (2013), no. 1, 50-64. https://doi.org/10.1080/10652469.2012.665910
  27. Y. Watanabe, Zum Riemanschen binomischen Lehrsatz, Proc. Phys. Math. Soc. Japan 14 (1932), 22-35.