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A Multitude of Expressions for the Stirling Numbers of the First Kind

Jacob Katriel
  • Schulich Faculty of Chemistry, Technion – Israel Institute of Technology, Haifa 32000, Israel. E-mail:
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Published Online: 2010-07-08 | DOI: https://doi.org/10.1515/integ.2010.023

Abstract

It is shown that the Stirling numbers of the first kind can be expressed in the form , where Q is a product of k – 1 linear factors in the indices j 1, j 2, . . . , j k–1 and α is a normalization coefficient determined by the condition . Several types of Q's are shown to yield Stirling numbers (“be Stirling”), and some more are conjectured to do so. The complete characterization of the set of Q's that are Stirling is not yet available. This set can be divided into subsets, within each of which different Q's are related by permutational symmetries. The case Q = j 1. j 2 . . . j k–1 is due to Adamchik (J. Comput. Appl. Math. 79: 119–130, 1997).

Keywords.: Stirling numbers of the first kind

About the article

Received: 2009-11-05

Revised: 2010-02-11

Accepted: 2010-02-22

Published Online: 2010-07-08

Published in Print: 2010-07-01


Citation Information: Integers, Volume 10, Issue 3, Pages 273–297, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integ.2010.023.

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