From the Rodrigues representation of polynomial eigenfunctions of a second order linear hypergeometric-type differential (difference or q-difference) operator, complementary polynomials for classical orthogonal polynomials are constructed using a straightforward method. Thus a generating function in a closed form is obtained. For the complementary polynomials we present a second order linear hy...

We obtain Mehler’s formula and the Rogers formula for the continuous big qHermite polynomials Hn(x; a|q). Instead of working with the polynomials Hn(x; a|q) directly, we consider the equivalent forms in terms of the bivariate Rogers-Szegö polynomials hn(x, y|q) recently introduced by Chen, Fu and Zhang. It turns out that Mehler’s formula for Hn(x; a|q) involves a 3φ2 sum, and the Rogers formula...

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator with polynomial coefficients. In this paper we present a study of classical orthogonal polynomials in a more general framework by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified...

In this paper, we deal with value distribution for q-shift polynomials of transcendental meromorphic functions with zero order and obtain some results which improve the previous theorems given by Liu and Qi [18]. In addition, we investigate value sharing for q-shift polynomials of transcendental entire functions with zero order and obtain some results which extend the recent theorem given by Li...

We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property in this paper. By giving q-varying variations, we identify the shape approximate roots polynomials, solution order, find conjectures associated with them. Furthermore, based on create Mandelbrot set Julia variety figures.

The purpose of this paper is to organize various types higher order q-differential equations that are connected q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the polynomials, we show symmetric order. Moreover, derive special for approximate roots which solutions equations.

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthe...

In this paper, we construct degenerate q-tangent numbers and polynomials determine their related properties. Based on these polynomials, also confirm that the structure of approximate root changes according to in q h. We find differential equations have as solutions other coefficients, confirming relationships among these.

The Hurwitz stable polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p, q ∈ R[x]: p(x) = anx n + an−1x n−1 + · · ·+ a1x + a0 q(x) = bmx m + bm−1x m−1 + · · ·+ b1x + b0 the Hadamard product (p ∗ q) is defined as (p ∗ q)(x) = akbkx + ak−1bk−1x + · ...