scholarly journals The dual of number sequences, Riordan polynomials, and Sheffer polynomials

2021 ◽  
Vol 10 (1) ◽  
pp. 153-165
Author(s):  
Tian-Xiao He ◽  
José L. Ramírez
Keyword(s):  
Catalan Numbers ◽  
Combinatorial Identities ◽  
Bell Numbers ◽  
Set Partitions ◽  
Polynomial Sequences ◽  
Number Sequences ◽  
Sheffer Polynomials ◽  
Sheffer Polynomial

Abstract In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

scholarly journals New Bell–Sheffer Polynomial Sets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 71 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Paolo Ricci
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Higher Order ◽  
Combinatorial Analysis ◽  
Bell Numbers ◽  
Integer Sequences ◽  
Shift Operators ◽  
Sheffer Polynomials ◽  
Logarithmic Functions ◽  
Sheffer Polynomial

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.

Bell–Sheffer exponential polynomials of the second kind

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pierpaolo Natalini ◽  
Paolo Emilio Ricci
Keyword(s):  
Higher Order ◽  
Bell Numbers ◽  
Integer Sequences ◽  
Exponential Polynomials ◽  
Sheffer Polynomials

AbstractIn recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers and several integer sequences related to them have been studied. In the present paper, new sets of Bell–Sheffer polynomials are introduced. Connections with Bell numbers are shown.

An algebraic approach to Sheffer polynomial sequences

2013 ◽  
Vol 25 (4) ◽  
pp. 295-311 ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Elisabetta Longo
Keyword(s):  
Algebraic Approach ◽  
Polynomial Sequences ◽  
Sheffer Polynomial

scholarly journals Identities for linear recursive sequences of order $ 2 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
General Rule ◽  
Transformation Techniques ◽  
Polynomial Sequences ◽  
Recursive Sequences ◽  
Number Sequences ◽  
Sequence Transformation

<p style='text-indent:20px;'>We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [<xref ref-type="bibr" rid="b16">16</xref>]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown.</p>

scholarly journals On Sheffer polynomial families

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 4
Author(s):  
Sandra Pinelas ◽  
Paolo Emilio Ricci
Keyword(s):  
Differential Equations ◽  
Exponential Functions ◽  
Sheffer Polynomials ◽  
Fractional Type ◽  
Sheffer Polynomial

Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.

scholarly journals On multivariable cumulant polynomial sequences with applications

2016 ◽  
Vol 7 (1) ◽  
Author(s):  
Elvira Di Nardo
Keyword(s):  
Main Tool ◽  
Multivariate Functions ◽  
Random Sums ◽  
Open Problems ◽  
Polynomial Sequence ◽  
Polynomial Sequences ◽  
Parameter Estimations ◽  
New Family ◽  
Random Index ◽  
Sheffer Polynomial

A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment se- quences can be recovered as well. The main tool is a formal generalization of random sums,  when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L\'evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper.

scholarly journals Pattern-Avoiding Set Partitions and Catalan Numbers

10.37236/2048 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Kernel Method ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Narayana Numbers ◽  
Combinatorial Methods

We identify several subsets of the partitions of $[n]$, each characterized by the avoidance of a pair of patterns, respectively of lengths four and five.  Each of the classes we consider is enumerated by the Catalan numbers.  Furthermore, the members of each class having a prescribed number of blocks are enumerated by the Narayana numbers.  We use both algebraic and combinatorial methods to establish our results.  In some of the cases, we make use of the kernel method to solve the recurrence arising when a further statistic is considered.  In other cases, we define bijections with previously enumerated classes which preserve the number of blocks.  Two of our bijections are of an algorithmic nature and systematically replace the occurrences of one pattern with those of another having the same length.

scholarly journals Combinatorial identities for restricted set partitions

2016 ◽  
Vol 339 (4) ◽  
pp. 1306-1314 ◽  
Author(s):  
Augustine O. Munagi
Keyword(s):  
Combinatorial Identities ◽  
Set Partitions

scholarly journals A Geometric Interpretation of the Intertwining Number

10.37236/7986 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Mahir Bilen Can ◽  
Yonah Cherniavsky ◽  
Martin Rubey
Keyword(s):  
Algebraic Geometry ◽  
Stirling Numbers ◽  
Geometric Interpretation ◽  
Bell Numbers ◽  
Set Partitions ◽  
Depth Index ◽  
Borel Orbits ◽  
Algebraic Monoid

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.

scholarly journals Catalan numbers and pattern restricted set partitions

2012 ◽  
Vol 312 (20) ◽  
pp. 2979-2991
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Catalan Numbers ◽  
Set Partitions
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