OICC Press Mathematical Analysis and its Contemporary Applications 3 1 2021 01 01 Analytic differenceability of functions 10.30495/maca.2021.680048 EN Soodeh Mehboodi Zand Institute of Higher Education, Shiraz, Iran Mohammad Hadi Hooshmand Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran Journal Article 2021 01 01 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).