scholarly journals On a Family of Multivariate Modified Humbert Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Rabia Aktaş ◽  
Esra Erkuş-Duman
Keyword(s):  
Generating Functions ◽  
Special Cases ◽  
Humbert Polynomials

This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable 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 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.

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.

scholarly journals New Generalizations of Exponential Distribution with Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
N. A. Rather ◽  
T. A. Rather
Keyword(s):  
Exponential Distribution ◽  
Generating Functions ◽  
Weibull Distributions ◽  
Generalized Exponential Distribution ◽  
Special Cases ◽  
Moment Generating Functions

The main purpose of this paper is to present k-Generalized Exponential Distribution which among other things includes Generalized Exponential and Weibull Distributions as special cases. Besides, we also obtain three-parameter extension of Generalized Exponential Distribution. We shall also discuss moment generating functions (MGFs) of these newly introduced distributions.

ANISOTROPIC STEP, SURFACE CONTACT, AND AREA WEIGHTED DIRECTED WALKS ON THE TRIANGULAR LATTICE

2002 ◽  
Vol 16 (09) ◽  
pp. 1269-1299 ◽  
Author(s):  
A. C. OPPENHEIM ◽  
R. BRAK ◽  
A. L. OWCZAREK
Keyword(s):  
Triangular Lattice ◽  
Generating Functions ◽  
Continued Fractions ◽  
Functional Equations ◽  
Square Lattice ◽  
Combinatorial Objects ◽  
Special Cases ◽  
Explicit Formulae ◽  
Directed Walks ◽  
Marked Area

We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

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 On Dual Hyperbolic Generalized Fibonacci Numbers

2019 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals On Generalized Third-Order Pell Numbers

2019 ◽  
pp. 1-18
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Third Order ◽  
Pell Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we investigate the generalized third order Pell sequences and we deal with, in detail, three special cases which we call them third order Pell, third order Pell-Lucas and modified third order Pell 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.

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