scholarly journals The Translated Dowling Polynomials and Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mahid M. Mangontarum ◽  
Amila P. Macodi-Ringia ◽  
Normalah S. Abdulcarim
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Hankel Transform ◽  
Bell Polynomials ◽  
Basic Properties ◽  
Whitney Numbers ◽  
Exponential Generating Function

More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals A Note on the (p, q)-Hermite Polynomials

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Ayhan Esi ◽  
Serkan Araci
Keyword(s):  
Integral Representation ◽  
Recurrence Relation ◽  
Explicit Formula ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Hermite Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.

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 Miscellaneous Properties of the Gamma Distribution Polynomials

2019 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Gamma Distribution ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Special Polynomial

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

scholarly journals Hopping from Chebyshev Polynomials to Permutation Statistics

10.37236/8413 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Jordan O. Tirrell ◽  
Yan Zhuang
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Permutation Statistics ◽  
Domino Tilings ◽  
Peak Number ◽  
Exponential Generating Function

We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the (-1)-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue of this action due to Sun and Wang.

scholarly journals Consecutive patterns in permutations: clusters and generating functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde ◽  
Marc Noy
Keyword(s):  
Entire Function ◽  
Large Class ◽  
Generating Function ◽  
Generating Functions ◽  
Main Tool ◽  
Cluster Method ◽  
Exponential Generating Function ◽  
International Audience ◽  
Patterns In Permutations

International audience We use the cluster method in order to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. Our main tool is the cluster method of Goulden and Jackson. We also prove some that, for a large class of patterns, the inverse of the exponential generating function counting occurrences is an entire function, but we conjecture that it is not D-finite in general. On utilise la mèthode des clusters pour ènumèrer permutations qui èvitent motifs consècutifs. On redèmontre et on gènèralise d'une manière unifièe plusieurs rèsultats et on obtient de nouveaux rèsultats pour certains motifs de longueur 4 et 5, ainsi que pour certaines familles infinies de motifs. L'outil principal c'est la mèthode des clusters de Goulden et Jackson. On dèmontre aussi que, pour une grande classe de motifs, l'inverse de la sèrie gènèratrice exponentielle qui compte occurrences est une fonction entière, mais on conjecture qu'elle n'est pas D-finie en gènèral.

scholarly journals A q-analogue of α־-Whitney numbers

2018 ◽  
Vol 12 (1) ◽  
pp. 178-191 ◽  
Author(s):  
B.S. El-Desouky ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Explicit Formulas ◽  
Basic Properties ◽  
Whitney Numbers

We define the (q,??)-Whitney numbers which are reduced to the ??-Whitney numbers when q ? 1. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the ??-Whitney-Lah numbers as a generalization of the r-Whitney-Lah numbers and we introduce their important basic properties.

scholarly journals The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers

2019 ◽  
Vol 12 (3) ◽  
pp. 1122-1137
Author(s):  
Roberto Bagsarsa Corcino ◽  
Charles Montero ◽  
Maribeth Montero ◽  
Jay Ontolan
Keyword(s):  
Explicit Formula ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Whitney Numbers

This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms  of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established.  Moreover, a $q$-analogue of the explicit formula is obtained.

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