Polynomials Hn Research Articles

A criterion for the normality of polynomials over finite fields based on their coefficients

An irreducible polynomial over Fq is said to be normal over Fq if its roots are linearly independent over Fq. We show that there is a polynomial hn(X1,…,Xn)∈Z[X1,…,Xn], independent of q, such that if an irreducible polynomial f=Xn+a1Xn−1+⋯+an∈Fq[X] is such that hn(a1,…,an)≠0, then f is normal over Fq. The polynomial hn(X1,…,Xn) is computed explicitly for n≤5 and partially for n=6. When charFq=p, we also show that there is a polynomial hp,n(X1,…,Xn)∈Fp[X1,…,Xn], depending on p, which is simpler than hn but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.

Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium

Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.

A Quadruple Integral Involving the Hermite polynomial Hn(x): Derivation and Evaluation

A closed form expression of a quadruple integral involving the Hermite polynomial Hn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

New Integral Equations for the Monic Hermite Polynomials

In this article, we are study the question of existence of integral equation for the monic Hermite polynomials Hn, where the intervening real function does not depend on the index n, well-known by the linear functional Wx given by its moments Hn(x) = ⟨Wx, tn⟩, n ≥ 0, ♣x♣ < ∞. Also, we obtain some properties of the zeros of this intervening function. Furthermore, we obtain an integral representation of the Dirac mass δx, for every real number x.

A Quadruple Integral Involving the Hermite Polynomial Hn(x): Derivation and Evaluation

A closed form expression of a quadruple integral involving the Hermite polynomial $H_{n}(x)$ is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

New Generalized Apostol-Frobenius-Euler polynomials and their Matrix Approach

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials ℋn[m−1,α](x; c,a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix ????[m−1,α](x; c,a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix ????[m−1,α](c,a; λ; u), we deduce a product formula for ????[m−1,α](x; c,a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix ????[m−1,α](x; c,a; λ; u), which involving the generalized Pascal matrix.

Numerator polynomials of Riordan matrices

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials gn(x)=(1−x)n+1∑m=0∞pn(m)xm, where pn(m) is the polynomial of degree ≤n, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials hn(x)=(1−x)2n+1∑m=0∞(m+1)...(m+n)pn(m)xm are the numerator polynomials of the generating functions of diagonals of the exponential Riordan matrices. In paper, the properties of these two types of numerator polynomials and the constructive relationships between them are considered.

Generalized Horadam Polynomials and Numbers

Abstract We consider the polynomials hn,m(x) (m ≥ 2) and the numbers hn,m (x = 1), which are the generalized Horadam polynomials and the generalized Horadam numbers, respectively. We also consider the polynomials h(s) n,m(x)- convolutions of the polynomials hn,m(x), and the sequence of numbers h(s) n,m- convolutions of the numbers hn,m, where s ≥ 0.

A STUDY ON GENERALIZED HERMITE POLYNOMIALS

In this paper, we obtain generating functions involving hyper geometric functions. Rodrigues type formula of Hermite polynomials which is closely related to generalized Hermite polynomials of Dattoli et. al. These results provide useful extensions of the well known results of classical Hermite polynomials Hn(x).

Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the nth monic Hermite polynomial Hn converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof–based on the recursion satisfied by the Hermite polynomials–we show that the generating function of the moments of roots of Hn is convergent and it satisfies a fixed point equation, which is also satisfied by c(z2), where c(z) is the generating function of the Catalan numbers Ck. In the second proof we compute the leading and the second leading term of the kth moments (as a polynomial in n) of Hn and show that the first one coincides with Ck/2, the (k/2)th Catalan number, where k is even and the second one is given by −(22k−1−(2k−1k)). We also mention the known result that the expectation of the characteristic polynomial (pn) of a Wigner random matrix is exactly the Hermite polynomial (Hn), i.e. Epn(x)=Hn(x), which suggests the presence of a deep connection between the Hermite polynomials and Wigner matrices.

Integral representations and new generating functions of Chebyshev polynomials

In this paper we use the two-variable Hermite polynomials and their operational rules to derive integral representations of Chebyshev polynomials. The concepts and the formalism of the Hermite polynomials Hn(x;y) are a powerful tool to obtain most of the properties of the Chebyshev polynomials. By using these results, we also show how it is possible to introduce relevant generalizations of these classes of polynomials and we derive for them new identities and integral representations. In particular we state new generating functions for the first and second kind Chebyshev polynomials. 2000 AMS Classification: 33C45, 33D45.

Markov type polynomial inequality for some generalized Hermite weight

ABSTRACT In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

Lifting q-difference operators for Askey-Wilson polynomials and their weight function

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials Hn(x|q) of Rogers into the Askey-Wilson polynomials pn(x; a, b, c, d|q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form ɛq(cqDq) (where cq are some constants), defined as Exton’s q-exponential function ɛq(z) in terms of the Askey-Wilson divided q-difference operator Dq. We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsHn(x|q) up to the weight function, associated with the Askey-Wilson polynomials pn(x; a, b, c, d|q).

On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials

We construct a q-difference operator that lifts the continuous q-Hermite polynomials Hn(x| q) of Rogers up to the continuous big q-Hermite polynomials Hn(x; a| q) on the next level in the Askey scheme of basic hypergeometric polynomials. This operator is defined as Exton's q-exponential function εq(aqDq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator . We next show that one can move two steps more upwards in order first to reach the Al-Salam–Chihara family of polynomials Qn(x; a, b | q), and then the continuous dual q-Hahn polynomials pn(x; a, b, c| q). In both these cases, lifting operators, respectively, turn out to be convolution-type products of two and three one-parameter q-difference operators of the same type εq(aqDq) at the initial step. At each step, we also determine q-difference operators that lift the weight function for the continuous q-Hermite polynomials Hn(x| q) successively up to the weight functions for Hn(x; a| q), Qn(x; a, b | q) and pn(x; a, b, c| q).

On integral and finite Fourier transforms of continuous q-Hermite polynomials

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials Hn(x|q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.

On continuous q-Hermite polynomials and the classical Fourier transform

We prove that the classical Fourier-transform operator intertwines two q-difference equations for the continuous q-Hermite polynomials Hn(x|q) of Rogers, which are associated with the two distinct sets of values for the parameter q: 0 < q < 1 and 1 < q < ∞.

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f (x,y) and g(x,y) on D when n → 8, provided f (x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipMα(0 < α ≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipMα and f(x,y) belongs to C1(D), the inner product between the partial derivative of Hakopian interpolation polynomial \(\frac{\partial }{{\partial x}}H_n (f;x,y)\) and g(x,y) on D converges to that between \(\frac{\partial }{{\partial x}}f(x,y)\) and g(x,y) on D when n→∞.

The factorization of a q-difference equation for continuous q-Hermite polynomials

We argue that a customary q-difference equation for the continuous q-Hermite polynomials Hn(x|q) can be written in the factorized form as , where is some explicitly known q-difference operator. This means that the polynomials Hn(x|q) are in fact governed by the q-difference equation , which is simpler than the conventional one. It is shown that a similar factorization holds for the continuous q−1-Hermite polynomials hn(x|q).

The bivariate Rogers–Szegö polynomials

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers–Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3ϕ2 sum and the Rogers formula involves a 2ϕ1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics based on a q-extension of the classical Hermite polynomials H n (x) recently introduced by us in R. Alvarez-Nodarse et al.: Boletin de la Sociedad Matematica Mexicana (3) 8 (2002) 127. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line ℝ with respect to a positive weight function. A detailed description of the corresponding q-system is carried out.