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Generalized Stirling and Lah numbers

1996
*
Discrete Mathematics
*

Within this theory three

doi:10.1016/0012-365x(95)00112-a
fatcat:wpsntlgbcvddzlzkftaplug36m
*generalizations*of Stifling*numbers*of the second kind,*and*of*Lah**numbers*, are developed. ... The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces,*and*chains. ... Acknowledgements I am grateful to Reid Davis for assistance in the formulation*and*proof of Theorem 5.1. ...##
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Some Theorems on Tauber's Generalized Stirling, Lah and Bell Numbers

2019
*
European Journal of Pure and Applied Mathematics
*

In this paper, some properties for Tauber's

doi:10.29020/nybg.ejpam.v12i3.3424
fatcat:3x4h6g5iuzbp3fg6x5grxg545i
*generalized**Stirling**and**Lah**numbers*are obtained including other forms of recurrence relations, orthogonality*and*inverse relations, rational*generating*function ...*and*explicit formual in symmetric function form. ... The Bell*numbers*B h,n can be computed in terms of Tauber's*generalized**Lah**and**Stirling**numbers*of the second kind. ...##
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On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks
[article]

2017
*
arXiv
*
pre-print

Interestingly, the coefficients in individual expansions turn out to be

arXiv:1701.00600v1
fatcat:5lvjgwlfbffe7mkovxmgw2kco4
*generalizations*of the*Stirling**numbers*of the first kind*and*the*Lah**numbers*. ... This paper studies the*generalizations*of the*Stirling**numbers*of both kinds*and*the*Lah**numbers*in association with the normal order problem in the Weyl algebra W=〈 x,D|Dx-xD=1〉. ...*Stirling**numbers*of the first kind*and**Lah**numbers*for words. ...##
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On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

2017
unpublished

Interestingly, the coefficients in individual expansions turn out to be

fatcat:teylshf23jc67jk2fqj77c26ua
*generalizations*of the*Stirling**numbers*of the first kind*and*the*Lah**numbers*. ... This paper studies the*generalizations*of the*Stirling**numbers*of both kinds*and*the*Lah**numbers*in association with the normal ordering problem in the Weyl algebra W = x, D|Dx − xD = 1. ... Introduction The*Stirling**numbers*of both kinds*and*the*Lah**numbers*are ubiquitous in combinatorics. ...##
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EVOLUTIONARY MATHEMATICS AND SCIENCE FOR ULTIMATE GENERALIZATION OF LAH NUMBERS/(BINOMIAL COEFFICIENTS): SUMS/(ALTERNATE SUMS) OF ORTHOGONAL PRODUCTS OF STIRLING NUMBERS
[chapter]

2021

Evolutionary mathematics

doi:10.17613/xvkz-t375
fatcat:no3fler475chxk5qnwcvfr3m3e
*and*science for Ultimate*Generalization*of*Lah**Numbers*/(Binomial Coefficients): Sums/(Alternate Sums) of Orthogonal Products of*Stirling**Numbers*. ... STEAM-VOL3-NUM6-JUN2021; ISBN 978-0-9890870-3-2. ...............ABSTRACT: We first introduce*Stirling**and**Lah**numbers*via recursion*and*express*Lah**numbers**and*binomial coefficients as sums*and*alternate ... FOR ULTIMATE*GENERALIZATION*OF*LAH**NUMBERS*/(BINOMIAL COEFFICIENTS): SUMS/(ALTERNATE SUMS) OF ORTHOGONAL PRODUCTS OF*STIRLING**NUMBERS*" is Dr. ...##
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An Explicit Formula for the Bell Numbers in Terms of the Lah and Stirling Numbers

2015
*
Mediterranean Journal of Mathematics
*

In the paper, the author finds an explicit formula for computing Bell

doi:10.1007/s00009-015-0655-7
fatcat:64fy3ptyejgvjnib7zekdxnvle
*numbers*in terms of*Lah**numbers**and**Stirling**numbers*of the second kind. ... find a new explicit formula for computing Bell*numbers*B n in terms of*Lah**numbers*L(n, k)*and**Stirling**numbers*of the second kind S(n, k). (10) Proof. ... There are two kinds of*Stirling**numbers*:*Stirling**numbers*of the first*and*second kinds. ...##
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The r-Bell numbers and matrices containing non-central Stirling and Lah numbers

2019
*
Journal of Mathematics and Computer Science
*

Moreover, some matrix relations are obtained involving r-Bell

doi:10.22436/jmcs.019.03.05
fatcat:d23epglczfdo3gtnhobhqh2x7u
*numbers*, r-*Stirling**numbers*of the second kind, r-*Lah**numbers*, non-central*Stirling**numbers*of the second kind,*and*the ordinary*Lah**numbers*... The other formula is expressed in terms of the non-central*Stirling**numbers*of the second kind*and*the ordinary*Lah**numbers*. ... The authors would also like to thank the referees for their helpful comments*and*suggestions. ...##
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Analogies of the Qi formula for some Dowling type numbers
[article]

2018
*
arXiv
*
pre-print

In the paper, the authors establish explicit formulas for the Dowling

arXiv:1808.02663v1
fatcat:q4vutyxxirf2bnrquywvjuwddm
*numbers**and*their*generalizations*in terms of*generalizations*of the*Lah**numbers**and*the*Stirling**numbers*of the second kind. ... These results gen- eralize the Qi formula for the Bell*numbers*in terms of the*Lah**numbers**and*the*Stirling**numbers*of the second kind. ... The authors wish to thank the anonymous referee for reading the manuscript thoroughly which resulted in numerous corrections*and*improvements. ...##
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Associated Lah numbers and r-Stirling numbers
[article]

2014
*
arXiv
*
pre-print

An extension of the associated

arXiv:1404.5573v2
fatcat:jqyabdubgbfytbnxemh2yrxofa
*Stirling**and**Lah**numbers*to the r-*Stirling**and*r-*Lah**numbers*are also given. ... We introduce the associated*Lah**numbers*. Some recurrence relations*and*convolution identities are established. ... In section 9, we propose some*generating*functions of the s-associated r-*Stirling**numbers*. The s-associated*Lah**numbers*We start by introducing the s-associated*Lah**numbers*. ...##
###
Page 4059 of Mathematical Reviews Vol. 58, Issue 6
[page]

1979
*
Mathematical Reviews
*

Several variants of the inversion, single

*and*repeated, of linear relations involving*generalized**Stirling**and**Lah**numbers*are con- sidered. ... “In this paper we give a systematic description of the basic properties of*generalized**Stirling**and**Lah**numbers*,*and*derive some new formulas involving these*numbers*.” ...##
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Inversions relating Stirling, tanh, Lah numbers and an application to Mathematical Statistics
[article]

2004
*
arXiv
*
pre-print

Inversion formulas have been found, converting between

arXiv:math/0405594v1
fatcat:cj667n2d5rczhonjss5ryv4kue
*Stirling*, tanh*and**Lah**numbers*. ... New identities for*Stirling**and*tangent*numbers**and*polynomials have been derived from the*general*inverse relations. ... Since in the third*and*fourth inversion formulas*Stirling**numbers*convert between*Lah**and*tanh*numbers*,*and*in the fifth*and*sixth tanh*numbers*convert between*Lah**and**Stirling**numbers*, the proof is complete ...##
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Convolutions for Stirling numbers, Lah numbers, and binomial coefficients
[article]

2021
*
arXiv
*
pre-print

We present various identities in the form of convolutions involving

arXiv:2103.15644v1
fatcat:souhpsoucredtfxidihldkkht4
*Stirling**numbers*of both kinds,*Lah**numbers*,*and*binomial coefficients. Certain convolution polynomials are discussed also. ... In this article we carry out a systematic study of such convolutions for the double sequences of*Stirling**numbers*of the first*and*second kinds, ( , ) s n k*and*( , ) S n k , the*Lah**numbers*( , ) L n ... In this study we produce various identities relating the above*numbers*. Excellent reference books for the*Stirling**and**Lah**numbers*are the classical books of Comtet [15]*and*Graham et al. [17] . ...##
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Multi-Lah numbers and multi-Stirling numbers of the first kind

2021
*
Advances in Difference Equations
*

AbstractIn this paper, we introduce multi-

doi:10.1186/s13662-021-03568-6
fatcat:7eiexb5t65gjlotrh7wo3j7ws4
*Lah**numbers**and*multi-*Stirling**numbers*of the first kind*and*recall multi-Bernoulli*numbers*, all of whose*generating*functions are given with the help of multiple ... In more detail, we represent the multi-Bernoulli*numbers*in terms of the multi-*Stirling**numbers*of the first kind*and*vice versa,*and*the multi-*Lah**numbers*in terms of multi-*Stirling**numbers*. ... In more detail, we represent the multi-Bernoulli*numbers*in terms of the multi-*Stirling**numbers*of the first kind*and*vice versa,*and*the multi-*Lah**numbers*in terms of multi-*Stirling**numbers*. ...##
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Lah numbers and Laguerre polynomials of order negative one
[article]

2016
*
arXiv
*
pre-print

In this article we point out interesting connections among

arXiv:1612.02876v1
fatcat:ssncwpaa5rag5flflknaaw76sq
*Lah**numbers*, Laguerre polynomials of order negative one,*and*exponential polynomials. ...*and*(1.5). This will reveal interesting connections of*Lah**numbers*to Laguerre polynomials*and*also to*Stirling**numbers*. ... Let ( , ) s n k*and*( , ) S n k be the*Stirling**numbers*of the first kind*and*the second kind correspondingly (see [9] ). ...##
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Recurrences of Stirling and Lah numbers via second kind Bell polynomials

2020
*
Discrete Mathematics Letters
*

first

doaj:4cf44b296dd44e88bad93e21fd5e78b1
fatcat:go7lkoh5affznkbkclapbywbrq
*and*second kinds, for 1-associate*Stirling**numbers*of the second kind, for the*Lah**numbers*,*and*for the binomial coefficients. ... Keywords: recurrence; Bell polynomial of the second kind;*Stirling**number*; 1-associate*Stirling**number*of the second kind;*Lah**number*. ... Acknowledgment The authors are thankful to anonymous referees for their careful corrections, helpful suggestions,*and*valuable comments on the original version of this paper. ...
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