In this paper, we suggest a new technique which uses Lagrange polynomials to get derivative-free iterative methods for solving nonlinear equations. With the use of the proposed technique and Steffens on-like methods, a new optimal fourth-order method is derived. By using three-degree Lagrange polynomials with other two-step methods which are efficient optimal methods, eighth-order methods can b...

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have few order, which are mixed with numbers Moreover, investigate some symmetric by applying properties

By using p-adic q-deformed fermonic integral on Zp, we define multiple the twisted (h,q)-Euler numbers of order α and polynomials of order α. After we obtain the multiplication formulae for the multiple twisted (h,q)-Euler polynomials. Also the multiple alternating sum obtained at the twisted (h,q)-Euler polynomials and the twisted (h,q)-Euler numbers. Mathematics Subject Classification: 05A10,...

We introduce the concept of D-operators associated to a sequence of polynomials (pn)n and an algebra A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family (pn)n of ...

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the measure |x|γ(1 − x2)1/2dx is derived which is based on a “reversing property” of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalizati...

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynom...

In this paper we define a family of polynomials closely related to the modified R-polynomials of the symmetric group and begin work toward a classification of the polynomials by using a combinatorial interpretation involving subwords of the maximal element in the Bruhat order. The problem of determining the precise conditions which make one of these polynomials zero motivates our work. We state...

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, fea...

For a measure μ on R, the situation is more subtle. One can always orthogonalize the subspaces of polynomials of different total degree (so that one gets a family of pseudo-orthogonal polynomials). The most common approach is to work directly with these subspaces, without producing individual orthogonal polynomials; see, for example [DX01]. One can also further orthogonalize the polynomials of ...