HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR X−cX-c

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator $X-c$, where $c$ is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the $q$-Hermite (resp. Charlier) polynomial is the only $H_{\alpha,q}$-classical (resp. $\mathcal{S}_{\lambda}$-classical) orthogonal polynomial, where $H_{\alpha, q}:=X+\alpha H_q$ and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\

Truncated-exponential-based Frobenius–Euler polynomials

In this paper, we first introduce a new family of polynomials, which are called the truncated-exponential based Frobenius–Euler polynomials, based upon an exponential generating function. By making use of this exponential generating function, we obtain their several new properties and explicit summation formulas. Finally, we consider the truncated-exponential based Apostol-type Frobenius–Euler polynomials and their quasi-monomial properties

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

Hahn's Problem with Respect to Some Perturbations of the Raising Operator X−C

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator X−c, where c is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the q-Hermite (resp. Charlier) polynomial is the only Hα,q-classical (resp. Sλ-classical) orthogonal polynomial, where Hα,q:=X+αHq and Sλ:=(X+1)−λτ−1

BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

In this paper, we introduce the concept of the $\mathbb{B}_{\alpha}$-classical orthogonal polynomials, where $\mathbb{B}_{\alpha}$ is the raising operator $\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}$, with nonzero complex number $\alpha$ and $\mathbb{I}$ representing the identity operator. We show that the Bessel polynomials $B^{(\alpha)}_n(x),\ n\geq0$, where $\alpha\neq-{m}/{2}, \ m\geq -2, \ m\in \mathbb{Z}$, are the only $\mathbb{B}_{\alpha}$-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution

Symbolic approach to 2-orthogonal polynomial solutions of a third order differential equation

In a recent work, a generic differential operator on the vectorial space of polynomial functions was presented and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Considering a third order differential operator that does not increase the degree of polynomials, we search for polynomial eigenfunctions with the help of symbolic computations, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. Two examples are extensively described.4516-0A1C-E9CD | Teresa Augusta MesquitaN/