and Applied Analysis 3 see 8 . For 0 ≤ k ≤ n, derivatives of the nth degree modified q-Bernstein polynomials are polynomials of degree n − 1: d dx Bk,n ( x, q ) n ( qBk−1,n−1 ( x, q ) − q1−xBk,n−1 ( x, q )) ln q q − 1 1.9 see 8 . The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. In t...

In this paper, we present generalization of matching extensions in graphs and we derive combinatorial interpretation of wide classes of orthogonal and q-orthogonal polynomials. Specifically, we assign general weights to complete graphs, cycles and chains or paths defining matching extensions in these graphs. The generalized matching polynomials of these graphs have recurrences defining various ...

A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of...

ψ-extension of Gian-Carlo Rota's finite operator calculus due to Viskov [1, 2] is further developed. The extension relies on the notion of ∂ ψ-shift invariance and ∂ ψ-delta operators. Main statements of Rota's finite operator calculus are given their ψ-counterparts. This includes Sheffer ψ-polynomials properties and Rodrigues formula-among others. Such ψ-extended calculus delivers an elementar...

Kupershmidt and Tuenter have introduced reflection symmetries for the q-Bernoulli numbers and the Bernoulli polynomials in 2005 , 2001 , respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the cl...

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...

We investigate the homogeneous symmetric Macdonald polynomials Pλ(X; q, t) for the specialization t = q. We show an identity relying the polynomials Pλ(X; q, q) and Pλ “ 1−q 1−qkX; q, q k ” . As a consequence, we describe an operator whose eigenvalues characterize the polynomials Pλ(X; q, q). Résumé. Nous nous intéressons aux propriétés des polynômes de Macdonald symétriques Pλ(X; q, t) pour la...

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algori...

ABSTRACT: Knop and Sahi introduced a family of non-homogeneous and nonsymmetric polynomials, Gα(x; q, t), indexed by compositions. An explicit formula for the bivariate Knop-Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macd...

We propose an algorithm that calculates isogenies between elliptic curves defined over extension K of Q 2 . It consists in efficiently solving with a logarithmic loss 2-adic precision the first-order differential equation satisfied by isogeny. give some applications, especially computing finite fields characteristic and irreducible polynomials, both quasi-linear time degree.