scholarly journals The Hankel Transform ofq-Noncentral Bell Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino ◽  
Jay M. Ontolan ◽  
Charrymae M. Perez-Fernandez ◽  
Ednelyn R. Cantallopez
Keyword(s):  
Stirling Numbers ◽  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Bell Numbers

We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of theq-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, aq-analogue of noncentral Bell numbers is defined and its Hankel transform is established.

Hankel Transform of the Type 2 (p,q)-Analogue of r-Dowling Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 343
Author(s):  
Roberto Corcino ◽  
Mary Ann Ritzell Vega ◽  
Amerah Dibagulun
Keyword(s):  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Whitney Numbers

In this paper, type 2 (p,q)-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in deriving the Hankel transform of the type 2 (p,q)-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel transform of the type 2 (p,q)-analogue of the r-Dowling numbers are established.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

scholarly journals Hankel Transform of (q,r)-Dowling Numbers

2019 ◽  
Vol 12 (2) ◽  
pp. 279-293
Author(s):  
Roberto B. Corcino ◽  
Mary Joy Regidor Latayada ◽  
Mary Ann Ritzell P. Vega
Keyword(s):  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Whitney Numbers

In this paper, we establish certain combinatorial interpretation for $q$-analogue of $r$-Whitney numbers of the second kind defined by Corcino and Ca\~{n}ete in the context of $A$-tableaux. We derive convolution-type identities by making use of the combinatorics of $A$-tableaux. Finally, we define a $q$-analogue of $r$-Dowling numbers and obtain some necessary properties including its Hankel transform.

scholarly journals Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

scholarly journals A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

2014 ◽  
Vol 318 ◽  
pp. 53-57 ◽  
Author(s):  
Pietro Codara ◽  
Ottavio M. D’Antona ◽  
Pavol Hell
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Interpretation
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