Multiparameter Higher Order Daehee and Bernoulli Numbers and Polynomials

This paper gives a new generalization of higher order Daehee and Bernoulli numbers and polynomials. We define the multiparameter higher order Daehee numbers and polynomials of the first and second kind. Moreover, we derive some new results for these numbers and polynomials. The relations between these numbers and Stirling and Bernoulli numbers are obtained. Furthermore, some interesting special cases of the generalized higher order Daehee and Bernoulli numbers and polynomials are deduced.

Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques

In this paper, we introduce the study of the general form of stochastic Van der Pol equation (SVDP) under an external excitation described by Gaussian white noise. The study involves the use of Wiener-Chaos expansion technique (WCE) and Wiener-Hermite expansion (WHE) technique. The application of these techniques results in a system of deterministic differential equations (DDEs). The resulting DDEs are solved by the numerical techniques and compared with the results of Monte Carlo (MC) simulations. Also, we introduce a new formula that facilitates handling the cubic nonlinear term of van der Pol equations. The main results of this study are: 1) WCE technique is more accurate, programmable compared with WHE and for the same order, WCE consumes less time. 2) The number of Gaussian random variables (GRVs) is more effective than the order of expansion. 3) The agreement of the results with the MC simulations reflects the validity of the forms obtained through theorem 3.1.

A Family of Generalized Stirling Numbers of the First Kind

A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky?[1]?and Gould?[2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.

Some New Results on the Number of Paths

Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by ?Anm. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.

The Exponential Flexible Weibull Extension Distribution

This paper is devoted to study a new generalization of the flexible Weibull with three parameters. This model is referred to as the exponential flexible Weibull extension (EFWE) distribution which exhibits bathtub-shaped hazard rate function. Some statistical properties such as the mode, median, the moment, quantile function, the moment generating function and order statistics are discussed. Moreover, the maximum likelihood method for estimating the model parameters and the Fisher’s information matrix is given. Finally, the advantage of the EFWE distribution is concluded by an application using real data.

Flexible Reduced Logarithmic-Inverse Lomax Distribution with Application for Bladder Cancer

In this paper, a new method for adding parameters to a well-established distribution to obtain more flexible new families of distributions is applied to the inverse Lomax distribution (IFD). This method is known by the flexible reduced logarithmic-X family of distribution (FRL-X). The proposed distribution can be called a flexible reduced logarithmic-inverse Lomax distribution (FRL-IL). The statistical and reliability properties of the proposed models are studied including moments, order statistics, moment generating function, and quantile function. The estimation of the model parameters by maximum likelihood and the observed information matrix are also discussed. In order to assess the potential of the newly created distribution. The extended model is applied to real data and the results are given and compared to other models.

Generalized Power Akshaya Distribution and Its Applications

In this paper, a new two-parameter distribution called generalized power Akshaya distribution extended from Akshaya distribution is introduced. This distribution is proposed to model lifetime data. Statistical properties like density, hazard, survival and moments are derived. Two parameters estimation is introduced using maximum likelihood and Bayesian techniques. Finally, an application of real data and a simulation study are introduced to illustrate the usefulness of the proposed distribution.