Volume 30, 2024, Number 4 (Online First)

Volume 30 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 (Online First)

  • Volume opened: 1 November 2024
  • Status: In progress

Congratulations to our Editorial Board Members Prof. Taekyun Kim, Dr. Mladen Vassilev-Missana and Prof. Krassimir Atanassov for their anniversaries in 2024!
Editorial. Pages 663–664
Vassia Atanassova
Editorial (PDF, 583 Kb)

Algorithms for representing positive odd integers as the sum of arithmetic progressions
Original research paper. Pages 665–680
Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Michael R. Schwob, Rama Venkat
Full paper (PDF, 226 Kb) | Abstract

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

Quotients of sequences under the binomial convolution
Original research paper. Pages 681–690
Pentti Haukkanen
Full paper (PDF, 221 Kb) | Abstract

This paper gives expressions for the solution \{a(n)\} of the equation

    \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*}

where b(0)\ne 0, that is, of the equation a \circ b = c in a, where \circ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.

Comments are closed.