ICMS 2018 Session 3: Symbolic Summation and Integration

ICMS 2018: Home, Sessions

Organizer

• Christoph Koutschan (Austrian Academy of Sciences)

Aim and Scope

The symbolic evaluation of (definite and indefinite) sums and integrals is an important topic in symbolic computation, and it is widely appreciated by users of computer algebra systems. From highschool students, who solve their homework integrals with freely available and easy-to-use tools such as WolframAlpha, to physicists, who employ highly specialized algorithms and software packages for simplifying complicated multi-sums, they all take advantage of the power of computer algebra software. This session is dedicated to the presentation and discussion of new trends in the areas of symbolic summation and symbolic integration, with a special emphasis on newly developed software for evaluating / simplifying sums and integrals, either in an explicit / implicit form or numerically.

Submission Guidelines

1. If you would like to give a talk in this session of ICMS, please submit a short abstract by e-mail to the session organizer. The deadline is March 31, 2018.
2. [Optional] The conference proceedings will be published in the Lecture Notes in Computer Science series by Springer. If you would like your extended abstract to be included therein, please submit it no later than April 21, 2018 via EasyChair.

For more details, also concerning LaTeX style files, see the Submission Guidelines of ICMS.

Talks/Abstracts

(in alphabetical order)

• Additive Decompositions in Primitive Extensions

  Shaoshi Chen (Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, China)

  Presentation given at ICMS

  Abstract:
  In this talk, we present some recent results on symbolic integration. We extend the classical Ostrogradsky-Hermite reduction for rational functions to more general functions in primitive extensions of certain types. For an element $f$ in such an extension $K$, the extended reduction decomposes $f$ as the sum of a derivative in $K$ and another element $r$ such that $f$ has an antiderivative in $K$ if and only if $r=0$; and $f$ has an elementary antiderivative over $K$ if and only if $r$ is a linear combination of logarithmic derivatives over the constants when $K$ is a logarithmic extension. Moreover, $r$ is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D-finite. This is a joint work with Hao Du and Ziming Li.

• Bernoulli Symbol and Sum of Powers

  Lin Jiu (Dalhousie University, Halifax, Nova Scotia, Canada)

  Presentation given at ICMS

  Abstract:
  Bernoulli symbol inherits umbral calculus properties and is also endowed with solid probabilistic background. In this talk, we introduce a symbolic representation of r-fold harmonic sums at negative indices, i.e., multiple harmonic power sums. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This is joint work with Christophe Vignat.

• Asymptotic Expansions

  Devendra Kapadia (Wolfram Research, Champaign, Illinois, USA)

  Presentation given at ICMS

  Abstract:
  Asymptotic expansions provide a powerful alternative to the traditional exact and numerical methods for solving problems in a wide variety of fields. We will illustrate this approach using new functions for computing asymptotic expansions of integrals, sums, differential equations and difference equations in the Wolfram Language. The talk will include examples of asymptotic expansions for exponential and oscillatory integrals, holonomic and special functions, and combinatorial sequences.

• Computability of general integrals and integral transforms

  Oleg Marichev (Wolfram Research, Champaign, Illinois, USA)

  Presentation given at ICMS (Mathematica notebook)

  Abstract:
  The majority of integrals (and classical integral transforms) that have been evaluated in numerous publications can be generalized to integrals and transforms that include the MeijerG function, either in integrands or in kernels of integral transforms. In the case of indefinite integrals one can also consider integrals with factors of the form Rj[x]^aj, where Rj[x] are arbitrary rational functions. In the talk we classify such general integrals and transforms and demonstrate how to evaluate them in closed form.

• Proving and Conjecturing Bounds for some Floor Function Sums

  Elaine Wong (RICAM, Austrian Academy of Sciences, Linz, Austria)

  Presentation given at ICMS

  Abstract:
  In 1957, a certain kind of floor function sum was first introduced by Jacobsthal, with work followed by that of Carlitz, Grimson, and Tverberg. More recently, Onphaeng and Pongsriiam proved some of the sharp upper and lower bounds for these sums using a formulation of Tverberg. In this talk, we introduce concise formulas for these sums, which can then be used to prove some of the upper and lower bounds analytically. We make an attempt to complete analysis on this type of problem and illustrate how computer algebra can be used to conjecture the remaining lower and upper bounds. This is joint work with Thotsaporn Thanatipanonda.