© 2018

Summability Calculus

A Comprehensive Theory of Fractional Finite Sums


  • The first book in the literature, which is devoted to fractional finite sums as an object of study on its own right

  • Unifies many disparate historical results in the works of prominent mathematicians, such as Gregory, Euler, Boole, and Ramanujan, and shows how such results relate to each other in significant ways

  • Contains new materials and identities that have not been published before


Table of contents

  1. Front Matter
    Pages i-xiii
  2. Ibrahim M. Alabdulmohsin
    Pages 1-19
  3. Ibrahim M. Alabdulmohsin
    Pages 21-54
  4. Ibrahim M. Alabdulmohsin
    Pages 55-63
  5. Ibrahim M. Alabdulmohsin
    Pages 65-91
  6. Ibrahim M. Alabdulmohsin
    Pages 93-113
  7. Ibrahim M. Alabdulmohsin
    Pages 115-131
  8. Ibrahim M. Alabdulmohsin
    Pages 133-149
  9. Back Matter
    Pages 151-165

About this book


This book develops the foundations of "summability calculus", which is a comprehensive theory of fractional finite sums. It fills an important gap in the literature by unifying and extending disparate historical results. It also presents new material that has not been published before. Importantly, it shows how the study of fractional finite sums benefits from and contributes to many areas of mathematics, such as divergent series, numerical integration, approximation theory, asymptotic methods, special functions, series acceleration, Fourier analysis, the calculus of finite differences, and information theory. As such, it appeals to a wide audience of mathematicians whose interests include the study of special functions, summability theory, analytic number theory, series and sequences, approximation theory, asymptotic expansions, or numerical methods. Richly illustrated, it features chapter summaries, and includes numerous examples and exercises. The content is mostly developed from scratch using only undergraduate mathematics, such as calculus and linear algebra.   


Fractional Finite Sums Analytic Summability Theory Divergent Series Special Functions Finite Differences Riemann Zeta Function Numerical Integration Series Acceleration Matrix Summability Methods Asymptotic Expansions Euler-Maclaurin Summation Formula Bernoulli Numbers Euler-Mascheroni constant Gamma and Polygamma functions

Authors and affiliations

  1. 1.King Abdullah University of Science and TechnologyDhahranSaudi Arabia

About the authors

Ibrahim M. Alabdulmohsin is a postdoctoral fellow at the Department of Computer Science at the King Abdullah University of Science and Technology in Saudi Arabia.

Bibliographic information

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