10.30495/maca.2021.680048

Analytic differenceability of functions

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  • Soodeh Mehboodi1,
  • Mohammad Hadi Hooshmand2,
  1. Zand Institute of Higher Education, Shiraz, Iran
  2. Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran

Published in Issue 2021-01-01

How to Cite

Analytic differenceability of functions. (2021). Mathematical Analysis and Its Contemporary Applications, 3(1). https://doi.org/10.30495/maca.2021.680048

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Abstract

Analytic summability of functions was introduced by Hooshmand in 2016. He used Bernoulli numbers and polynomials Bn(z) to define analytic summability and related analytic summand functions. Since the Bernoulli and Euler polynomials have many similarities, so it motivated us to define differenceability and introduce analytic difference function of a complex or real function by utilizing the Euler numbers and polynomials En(z). Also, we prove some criteria for analytic differenceability of analytic functions. Moreover, we observe that the analytic difference function is indeed a series of the Euler polynomials and arrive at some series convergence tests for Euler polynomial series P∞ n=0 cnEn(z).

Keywords

  • Bernoulli and Euler polynomials,
  • Bernoulli and Euler numbers,
  • Analytic summability
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References

  1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Washington, D.C. 1964.
  2. H. Alzer, Sharp bounds for The Bernoulli numbers, Arch. Math., 74 (2000), 207-211.
  3. T. M. Apostol, Introduction to analytic number theory, Springer, 1976.
  4. E. Artin, The gamma function, Holt Rhinehart & Wilson, New York, 1964; transl. by M. Butler from Einfuhrung un der Theorie der Gamma fonktion, Teubner, Leipzig, 1931.
  5. Bruce C. Berndt, Ramanujans notebooks (part I), Cambridge University Press, 1940.
  6. H.-F. Ge, New sharp bounds for the Bernoulli numbers and refinement of Becker-Stark inequalities, J. App. Math., vol. 2012, Article ID 137507, 7 pages, 2012.
  7. M. H. Hooshmand, Analytic summability of real and complex functions, J. Contemp. Math. Anal., 5 (2016), 262-268.
  8. S. Mehboodi and M. H. Hooshmand, Upper bounds for analytic summand functions and related inequalities, Math. Slovaca., article in press.
  9. M. Muller and D. Schleicher, Fractional sums and Euler-like identities, Ramanujan J., 21(2) (2010), 123-143.
  10. J. Sandor and B. Crstici, Handbook of number theory II, Volume 2, Springer, 2004.
  11. R. J. Webster, Log-Convex Solutions to the Functional Equation f(x + 1) = g(x)f(x): Γ-Type functions, J. Math. Anal. Appl., 209 (1997), 605-623.