Further generalization of the extended Hurwitz-Lerch Zeta functions

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

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

Journal of Applied Probability ◽

10.1239/jap/1101840560 ◽

2004 ◽

Vol 41 (4) ◽

pp. 1157-1170 ◽

Cited By ~ 2

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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On a Certain Extension of the Hurwitz-Lerch Zeta Function

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.2478/awutm-2014-0017 ◽

2014 ◽

Vol 52 (2) ◽

Author(s):

Rakesh K. Parmar ◽

R. K. Raina

Keyword(s):

Zeta Function ◽

Probability Distributions ◽

Integral Representations ◽

Mellin Transforms ◽

Special Cases ◽

Lerch Zeta Function ◽

Generating Relations

AbstractOur purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

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Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions

Axioms ◽

10.3390/axioms8020063 ◽

2019 ◽

Vol 8 (2) ◽

pp. 63 ◽

Cited By ~ 2

Author(s):

Rekha Srivastava ◽

Humera Naaz ◽

Sabeena Kazi ◽

Asifa Tassaddiq

Keyword(s):

Fractional Derivatives ◽

Series Representation ◽

Zeta Functions ◽

Weyl Transform ◽

Special Cases ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Bose Einstein ◽

Relationship Of ◽

The Relationship

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from ( 0 < ℜ ( s ) < 1 ) to ( 0 < ℜ ( s ) < μ ) . This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz–Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose–Einstein and Fermi–Dirac functions with Apostol–Euler–Nörlund polynomials are established to prove new identities.

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

Journal of Applied Probability ◽

10.1017/s002190020002091x ◽

2004 ◽

Vol 41 (04) ◽

pp. 1157-1170 ◽

Cited By ~ 2

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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An extension of the generalized Hurwitz-Lerch Zeta function of two variables

Filomat ◽

10.2298/fil1701091c ◽

2017 ◽

Vol 31 (1) ◽

pp. 91-96 ◽

Cited By ~ 4

Author(s):

Junesang Choi ◽

Rakesh Parmar

Keyword(s):

Zeta Function ◽

Zeta Functions ◽

Summation Formula ◽

Integral Representations ◽

Contour Integral ◽

Functions Of Two Variables ◽

Special Cases ◽

Lerch Zeta Function ◽

Function Of Two Variables

The main object of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate such its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a summation formula and Mellin-Barnes type contour integral representations. We also consider some important special cases.

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Generating Functions for Some Hypergeometric Functions of Four Variables via Laplace Integral Representations

Journal of Function Spaces ◽

10.1155/2021/7638597 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Jihad Younis ◽

Maged Bin-Saad ◽

Ashish Verma

Keyword(s):

Generating Functions ◽

Hypergeometric Functions ◽

Essential Role ◽

Integral Representations ◽

Laplace Integral ◽

Special Cases ◽

Generating Relations

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.

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New family of Whitney numbers

Filomat ◽

10.2298/fil1702309e ◽

2017 ◽

Vol 31 (2) ◽

pp. 309-320 ◽

Cited By ~ 2

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.

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Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽

10.3390/math8112023 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2023

Author(s):

Christopher Nicholas Angstmann ◽

Byron Alexander Jacobs ◽

Bruce Ian Henry ◽

Zhuang Xu

Keyword(s):

Initial Value Problems ◽

Fractional Derivatives ◽

Evolution Equations ◽

Integral Transform ◽

Initial Value ◽

Intrinsic Nature ◽

Recent Interest ◽

Special Cases ◽

Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

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A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽

10.3390/math9090979 ◽

2021 ◽

Vol 9 (9) ◽

pp. 979

Author(s):

Sandeep Kumar ◽

Rajesh K. Pandey ◽

H. M. Srivastava ◽

G. N. Singh

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Collocation Method ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Special Cases ◽

Collocation Approach ◽

Linear And Nonlinear ◽

Simulation Results ◽

Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

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