Related papers: A note on Hermite polynomials
We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first…
Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], we show how to compute the Hermite form H of A, and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in terms of…
It is well recognized that new types of exact travelling wave solutions to nonlinear partial differential equations can be obtained by modifications of the methods which are in hand. In this study, we extend the class of auxiliary equations…
Some $q-$analogues of the normal ordering of the operator $(X+sD)^n$ on the polynomials are derived.
This overview article gives an elementary approach to continuous q-Hermite polynomials. We stress their relation to Fibonacci, Lucas and Chebyshev polynomials and to some q-analogues of these polynomials.
In this work, we give the power series solutions around an ordinary point, in the case of variable coefficients, homogeneous sequential linear conformable fractional differential equations of order 2\alpha. Further, we introduce the…
The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
Motivated by classical results of approximation theory, we define an Hermite-type interpolation in terms of $n$-dimensional subspaces of the space of $n$ times continuously differentiable functions. In the main result of this paper, we…
For use in calculating higher-order coherent- and squeezed- state quantities, we derive generalized generating functions for the Hermite polynomials. They are given by $\sum_{n=0}^{\infty}z^{jn+k}H_{jn+k}(x)/(jn+k)!$, for arbitrary integers…
In this article we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two variable Kampe de Feriet family. We…
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and…
We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial,…
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
In this paper, we establish several new convex dominated functions and then we obtain new Hadamard type inequalities.
We analyze the Hermite polynomials $H_{n}(x)$ and their zeros asymptotically, as $n\to\infty.$ We obtain asymptotic approximations from the differential-difference equation which they satisfy, using the ray method. We give numerical…
We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish…
We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…
In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.