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Generalized Stirling Numbers, Convolution Formulae and p, q-Analogues

Published online by Cambridge University Press:  20 November 2018

Anne de Médicis  and
Pierre Leroux
Anne de Médicis
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota U.S.A.
Pierre Leroux
Affiliation:
LAC, IM Université du Québec à Montréal, Montréal, Québec
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Abstract

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In this paper, we study two generalizations of the Stirling numbers of the first and second kinds, inspired from their combinatorial interpretation in terms of 0-1 tableaux. They are the 𝔄-Stirling numbers and the partial Stirling numbers. In particular, we give a q and a p, q-analogue of convolution formulae for Stirling numbers of the second kind, due to Chen and Verde-Star, and we extend these formulae to Stirling numbers of the first kind. Included in this study are the a, d-progressive Stirlingnumbers, corresponding to 0-1 tableaux with column lengths from an arithmetic progression ﹛a + id﹜i≥0.

Résumé

Résumé

Dans cet article, nous étudions deux généralisations des nombres de Stirling de première et deuxième espèces, inspirées par leur interprétation combinatoire en termes de tableaux 0-1. Il s'agit des 𝔄-nombres de Stirlinget des nombres de Stirlingpartiels.Nous donnons en particulier des qet p, q-analogues de formules de convolution des nombres de Stirling de deuxième espèce, dues à Chen et Verde-Star, et nous étendons ces formules aux nombres de Stirling de première espèce. Les nombres de Stirlinga, d-progressifs,correspondant aux tableaux 0-1 dont les longueurs des colonnes font partie d'une progression arithmétique ﹛a + id﹜i≥0, sont également inclus dans cette étude.

Keywords

05A1511B6511B73
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 474 - 499
Copyright
Copyright © Canadian Mathematical Society 1995

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