Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...

Let i be a natural number and let x be a real number. Then i 7→ x is a finite sequence of elements of R. Let x be a real number and let n be a natural number. The functor xn is defined by: (Def. 1) xn = ∏(n 7→ x). Let x be a real number and let n be a natural number. Observe that xn is real. Let x be a real number and let n be a natural number. Then xn is a real number. We now state several pro...

Two factors having the same set of levels are said to be homologous. This paper aims to extend the domain of factorial models to designs that include homologous factors. In doing so, it is necessary rst to identify the characteristic property of those vector spaces that constitute the standard factorial models. We argue here that essentially every interesting statistical model speciied by a vec...

in this study a numerical method is developed to solve the hammerstein integral equations. to this end the kernel has been approximated using the leastsquares approximation schemes based on legender-bernstein basis. the legender polynomials are orthogonal and these properties improve the accuracy of the approximations. also the nonlinear unknown function has been approximated by using the berns...

in this study, the bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. to this aim, the operational matrices of integration and the product for bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. some examples are presented to illustrate the efficien...

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

Two-level factorial and fractional factorial designs have played a prominent role in the theory and practice of experimental design. Though commonly used in industrial experiments to identify the significant effects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. Instead, restrictions are imposed on the randomi...