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SHIFTING POWERS IN SPIVEY'S BELL NUMBER FORMULA.

Toufik Mansour1, Reza Rastegar2, Alexander Roitershtein3

  • 1Department of Mathematics, University of Haifa, 3498838 Haifa, Israel.

Quaestiones Mathematicae : Journal of the South African Mathematical Society
|May 2, 2022
PubMed
Summary

This study extends Spivey's Bell number formula to r-Whitney numbers, uncovering new identities using algebraic and combinatorial methods. These findings offer novel approaches to deriving existing formulas and introduce related results for r-Lah numbers.

Keywords:
Bell numberSpivey’s formulaWhitney numbercombinatorial identity

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Area of Science:

  • Combinatorics
  • Number Theory
  • Discrete Mathematics

Background:

  • Spivey's Bell number formula is a key result in combinatorics.
  • The r-Whitney numbers of the second kind generalize combinatorial sequences.
  • Understanding these numbers is crucial for various mathematical applications.

Purpose of the Study:

  • To extend Spivey's Bell number formula.
  • To explore new identities for r-Whitney numbers of the second kind.
  • To establish analogous results for r-Lah numbers.

Main Methods:

  • Algebraic manipulations involving infinite series and Dobinski-like formulas.
  • Combinatorial arguments analyzing statistics on set partitions.
  • Generalization of polynomial arguments in combinatorial identities.

Main Results:

  • New identities for r-Whitney numbers of the second kind were derived.
  • Novel methods for deducing Spivey's formula for W(n, k) were established.
  • An analogous identity for r-Lah numbers was proven.

Conclusions:

  • The study successfully generalized Spivey's Bell number formula.
  • Both algebraic and combinatorial approaches yielded significant results.
  • The findings contribute to the broader understanding of combinatorial numbers and their identities.