Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers

2012 ◽  
Vol 10 (1) ◽  
pp. 57-72 ◽  
Author(s):  
Nenad P. Cakić ◽  
Beih S. El-Desouky ◽  
Gradimir V. Milovanović
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Explicit Formulas ◽  
Generalized Stirling Numbers

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals Labeled Factorization of Integers

10.37236/139 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Augustine O. Munagi
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
New Technique ◽  
Combinatorial Interpretation ◽  
Prime Factors ◽  
Generalized Stirling Numbers ◽  
A New Technique ◽  
Explicit Enumeration ◽  
Factorization Of Integers

The labeled factorizations of a positive integer $n$ are obtained as a completion of the set of ordered factorizations of $n$. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of $n$. Our results include explicit enumeration formulas and some combinatorial identities. It is proved that labeled factorizations of $n$ are equinumerous with the systems of complementing subsets of $\{0,1,\dots,n-1\}$. We also give a new combinatorial interpretation of a class of generalized Stirling numbers.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Unimodality properties of generalized Stirling numbers

2015 ◽  
Vol 45 (9) ◽  
pp. 1583-1586
Author(s):  
Yi WANG ◽  
BaoXuan ZHU ◽  
Lily Li LIU
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals On a Class of Combinatorial Sums Involving Generalized Factorials

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
W.-S. Chou ◽  
L. C. Hsu ◽  
P. J.-S. Shiue
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Sums ◽  
Generalized Stirling Numbers

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

The multiset partitions and the generalized Stirling numbers

2020 ◽  
Vol 31 (5-6) ◽  
pp. 813-831
Author(s):  
Miloud Mihoubi ◽  
Asmaa Rahim ◽  
Said Taharbouchet
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals Modified approach to generalized Stirling numbers via differential operators

2010 ◽  
Vol 23 (1) ◽  
pp. 115-120 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad P. Cakić ◽  
Toufik Mansour
Keyword(s):  
Differential Operators ◽  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals A Newton Interpolation Approach to Generalized Stirling Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression ◽  
Recurrence Relation ◽  
Generating Function ◽  
Stirling Numbers ◽  
Divided Differences ◽  
Newton Interpolation ◽  
Basic Properties ◽  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

scholarly journals Lagrange and Wilson theorems for the generalized Stirling numbers

1938 ◽  
Vol 5 (4) ◽  
pp. 171-173 ◽  
Author(s):  
E. T. Bell
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers ◽  
Image Position

If m, n are integers, m > 0, n > 1, the generalized Stirling numbers are defined by the identity in x,

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