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 The Zeta-G Class: Some Computational and Analytical Aspects

2020 ◽  
Vol 42 ◽  
pp. e111
Author(s):  
Ana Carla Percontini ◽  
Frank Gomes-Silva ◽  
Gauss Moutinho Crdeiro ◽  
Pedro Rafael Marinho
Keyword(s):  
Maximum Likelihood ◽  
Generating Function ◽  
Real Data ◽  
Data Set ◽  
R Language ◽  
New Class ◽  
Mathematical Properties ◽  
Special Cases ◽  
Computational Aspects

We define a new class of distributions with one extra shapeparameter including some special cases. We provide numerical and computational aspects of the new class. We proposefunctions using the \textsf{R} language to fit any distribution in this family to a data set. In addition, such functions are implemented efficientlyusing the library \textsf{Rcpp} that enables the incorporation of the codes \textsf{C++} in \textsf{R} automatically. Some examples are presentedfor using the implemented routines in practice. We derive some mathematical properties of this class including explicit expressionsfor the moments, generating function and mean deviations. We discuss the estimation of the model parametersby maximum likelihood and provide an application to a real data set.

scholarly journals Identities of Symmetry for Generalized Euler Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Euler Polynomials ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the -adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.

scholarly journals Some New Classes of Generalized Hermite-Based Apostol-Euler and Apostol-Genocchi Polynomials

2015 ◽  
Vol 55 (1) ◽  
pp. 153-170 ◽  
Author(s):  
M. A. Pathan ◽  
Waseem A. Khan
Keyword(s):  
Generating Functions ◽  
Euler Polynomials ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae

Abstract In this paper, we introduce a new class of generalized Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials and derive some implicit summation formulae by applying the generating functions. These results extend some known summations and identities of generalized Hermite-Euler polynomials studied by Dattoli et al, Kurt and Pathan.

scholarly journals A new class of Appell-type Changhee-Euler polynomials and related properties

2021 ◽  
Vol 6 (12) ◽  
pp. 13566-13579
Author(s):  
Tabinda Nahid ◽  
◽  
Mohd Saif ◽  
Serkan Araci ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Graphical Representations ◽  
Euler Polynomials ◽  
New Class ◽  
New Type ◽  
Changhee Polynomials

<abstract><p>A remarkably large number of polynomials and their extensions have been presented and studied. In the present paper, we introduce the new type of generating function of Appell-type Changhee-Euler polynomials by combining the Appell-type Changhee polynomials and Euler polynomials and the numbers corresponding to these polynomials are also investigated. Certain relations and identities involving these polynomials are established. Further, the differential equations arising from the generating function of the Appell-type Changhee-Euler polynomials are derived. Also, the graphical representations of the zeros of these polynomials are explored for different values of indices.</p></abstract>

scholarly journals Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Yuan He ◽  
Chunping Wang
Keyword(s):  
Generating Function ◽  
Euler Polynomials ◽  
Special Cases

Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials are established by applying the generating function methods and some summation transform techniques, and various known results are derived as special cases.

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 Note on Type 2 Degenerate q-Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 681 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Sung-Soo Pyo
Keyword(s):  
Generating Function ◽  
Euler Polynomials ◽  
Integer Power ◽  
Power Sums

Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate q-Euler polynomials, which are derived from the fermionic p-adic q-integrals on Z p , and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate q-Euler polynomials arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously.

scholarly journals Some New Classes of Generalized Lagrange-Based Apostol Type Hermite Polynomials

2018 ◽  
Author(s):  
Waseem A. Khan ◽  
K.S. Nisar
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Euler Polynomials ◽  
General Symmetry ◽  
Basic Properties ◽  
Genocchi Polynomials ◽  
General Family ◽  
Summation Formulae ◽  
Explicit Representations

In this paper, we introduce a general family of Lagrange-based Apostol-type Hermite polynomials thereby unifying the Lagrange-based Apostol Hermite-Bernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We also define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

scholarly journals Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

scholarly journals A New Class of Estimators Based on a General Relative Loss Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1138
Author(s):  
Tao Hu ◽  
Baosheng Liang
Keyword(s):  
Loss Function ◽  
Asymptotically Normal ◽  
New Class ◽  
Statistical Inferences ◽  
Relative Loss ◽  
Special Cases ◽  
Class Of Estimators ◽  
Box Cox Transformation ◽  
Loss Estimator ◽  
Prostate Cancer Study

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

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