scholarly journals Multiplication formulas for q-Appell polynomials and the multiple q-power sums

2016 ◽  
Vol 70 (1) ◽  
pp. 1
Author(s):  
Thomas Ernst
Keyword(s):  
Generating Functions ◽  
Rational Numbers ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Power Sums ◽  
Special Cases

In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli  and Apostol-Euler  polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.

scholarly journals A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 652
Author(s):  
Serkan Araci ◽  
Mumtaz Riyasat ◽  
Shahid Wani ◽  
Subuhi Khan
Keyword(s):  
Generating Function ◽  
Zeta Functions ◽  
Euler Polynomials ◽  
Power Sums ◽  
New Class ◽  
Summation Formulae ◽  
Special Cases ◽  
Hybrid Class

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.

scholarly journals Properties of Partially Degenerate Complex Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1508
Author(s):  
Dojin Kim ◽  
Sangil Kim
Keyword(s):  
Differential Equations ◽  
Exponential Type ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Polynomial Sequences ◽  
General Properties ◽  
Symmetric Identities ◽  
Degenerate Version

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type 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 Certain results of hybrid families of special polynomials associated with appell sequences

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3833-3844 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi
Keyword(s):  
Generating Functions ◽  
Mixed Type ◽  
Bernoulli Polynomials ◽  
Operational Method ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
Appell Sequences

In this article, the Legendre-Gould-Hopper polynomials are combined with Appell sequences to introduce certain mixed type special polynomials by using operational method. The generating functions, determinant definitions and certain other properties of Legendre-Gould-Hopper based Appell polynomials are derived. Operational rules providing connections between these formulae and known special polynomials are established. The 2-variable Hermite Kamp? de F?riet based Bernoulli polynomials are considered as an member of Legendre-Gould-Hopper based Appell family and certain results for this member are also obtained.

scholarly journals On a Positivity Conjecture in the Character Table of $S_n$

10.37236/8186 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Sheila Sundaram
Keyword(s):  
Lexicographic Order ◽  
Character Table ◽  
Power Sums ◽  
Special Cases ◽  
Schur Positive

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $\mu\in [(n-4,1^4), (n)]$  or $\mu\in [(1^n), (3,1^{n-3})], $ or $\mu=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$  Many new Schur positivity questions are presented.

scholarly journals On Generalized Grahaml Numbers

2020 ◽  
pp. 42-57
Author(s):  
Yuksel Soykan
Keyword(s):  
Generating Functions ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized Grahaml sequences and we deal with, in detail, three special cases which we call them Grahaml, Grahaml-Lucas and modified Grahaml sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

2018 ◽  
Vol 6 (1) ◽  
pp. 69-84
Author(s):  
Jia Xu ◽  
Liwei Liu ◽  
Taozeng Zhu
Keyword(s):  
Generating Functions ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Queueing System ◽  
System Size ◽  
Heterogeneous Servers ◽  
Modified Bessel Functions ◽  
Multiple Vacations ◽  
Special Cases ◽  
Mean And Variance

AbstractWe consider anM/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at timetby employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

scholarly journals On Degenerate Truncated Special Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 144 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Exponential Polynomials ◽  
Special Polynomials ◽  
Summation Formulas ◽  
Special Cases ◽  
Proof Techniques

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

scholarly journals P-Partitions and p-Positivity

2019 ◽  
Author(s):  
Per Alexandersson ◽  
Robin Sulzgruber
Keyword(s):  
Linear Combination ◽  
Generating Functions ◽  
Quasisymmetric Functions ◽  
Vertical Strip ◽  
Power Sums ◽  
Llt Polynomials ◽  
Tutte Polynomials

AbstractUsing the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently, any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application, we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials, and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

M/GI/1 queues with services of both positive and negative customers

2004 ◽  
Vol 41 (4) ◽  
pp. 1157-1170 ◽  
Author(s):  
Yijun Zhu ◽  
Zhe George Zhang
Keyword(s):  
Generating Functions ◽  
Time Distribution ◽  
Opposite Sign ◽  
Probability Distributions ◽  
Waiting Time Distribution ◽  
Negative Customers ◽  
Special Cases ◽  
Stationary Waiting Time ◽  
Original Concept ◽  
Negative Arrivals

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.

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