On the differences of the generalized factorials at an arbitrary point and their combinatorial applications
type: Item
instance of: scholarly article
title: On the differences of the generalized factorials at an arbitrary point and their combinatorial applications
zbMATH Open document ID: 0539.05009
DOI: 10.1016/0012-365X(83)90089-4
published in: Discrete Mathematics
publication date: 1983-01-01T00:00:00Z
review text: In the paper there are defined and systematically investigated the Gould- Hopper numbers G(n,k;r,s) useful in biology and reliability. First of all there are given the definitions and notations, e.g. \(G(m,n;r,s)=[1/n! \Delta^ n(rx)_ m]_{x=s/r}=[1/n! \Delta^ n(rx+s)_ m]_{x=0}\) otherwise we can define \(G(m,n;r,s)=1/n!\sum^{n}_{k=0}(-1)^{n- k}\left( \begin{matrix} n\\ k\end{matrix} \right)(rk+s)_ m.\) This expression leads to the combinatorial interpretations of the numbers G(m,n;r,s) when both r and s are positive or negative integers. Further are given the recurrence relation and generating function especially the exponential generating function, then the connection with other numbers (e.g. with Stirling numbers) and limiting expressions and the orthogonality relations and other properties. Finally there are given the combinatorial and statistical applications (e.g. sequential occupancy, coupons collector problem and convolution of complete and zero-truncated general binomial distributions).
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 3857120
zbMATH Keywords: Gould-Hopper numbers
zbMATH Keywords: combinatorial interpretations
zbMATH Keywords: recurrence relation
zbMATH Keywords: generating function
author: Charalambos A. Charalambides
author: Markos V. Koutras
reviewed by: A.Huťa jun
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3857120