Classical Hermite Polynomials Research Articles

A New Extension of Combination of Linear and Bilinear Generating Relations in Function Spaces Associated with Hypergeometric Polynomials; Some New Applications of Two Signals in Special Functions

For a certain class of generalized hypergeometric polynomials, the first derive some special cases on linear and bilinear generating functions and then apply these generating functions in order to reduce the corresponding results for the classical Jacobi, Hermite, Laguerre and Gegenbauer Polynomials, hypergeometric functions of Gauss and functions of Bessel and Kelvin. They also consider several linear generating functions for these polynomials as well as for some multivariable Jacobi and multivariable Laguerre polynomials which were investigated in recent years. Some of the linear and bilinear generating functions, presented in this paper, are associated with the hypergeometric polynomials.

On generalized Hermite polynomials

<p>This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs.</p>

Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials

<abstract><p>In this study, we develop various features in special polynomials using the principle of monomiality, operational formalism, and other qualities. By utilizing the monomiality principle, new outcomes can be achieved while staying consistent with past knowledge. Furthermore, an explicit form satisfied by these polynomials is also derived. The emphasis of this study is to introduce the degenerate multidimensional Hermite polynomials (DMVHP) denoted as $ \mathbb{H}^{[r]}_n(j_1, j_2, j_3, \cdots, j_r; \vartheta) $, which are closely related to the classical Hermite polynomials and are a significant class of orthogonal polynomials. The fundamental properties, such as symmetric identities for these polynomials are also established. An operational framework is also established for these polynomials.</p></abstract>

Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

We find and discuss asymptotic formulas for orthonormal polynomials [Formula: see text] with recurrence coefficients [Formula: see text]. Our main goal is to consider the case where off-diagonal elements [Formula: see text] as [Formula: see text]. Formulas obtained are essentially different for relatively small and large diagonal elements [Formula: see text]. Our analysis is intimately linked with spectral theory of Jacobi operators [Formula: see text] with coefficients [Formula: see text] and a study of the corresponding second order difference equations. We introduce the Jost solutions [Formula: see text], [Formula: see text], of such equations by a condition for [Formula: see text] and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions [Formula: see text] by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for [Formula: see text] as [Formula: see text] in terms of the Wronskian of the solutions [Formula: see text] and [Formula: see text]. The formulas obtained for [Formula: see text] generalize the asymptotic formulas for the classical Hermite polynomials where [Formula: see text] and [Formula: see text]. The spectral structure of Jacobi operators [Formula: see text] depends crucially on a rate of growth of the off-diagonal elements [Formula: see text] as [Formula: see text]. If the Carleman condition is satisfied, which, roughly speaking, means that [Formula: see text], and the diagonal elements [Formula: see text] are small compared to [Formula: see text], then [Formula: see text] has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values [Formula: see text] of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of [Formula: see text] is discrete. We also review the case of stabilizing recurrence coefficients when [Formula: see text] tend to a positive constant and [Formula: see text] as [Formula: see text]. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.

Discrete weierstrass transform in discrete hermitian clifford analysis

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L2−space L2(R,exp(−x2/2)) to the Fock space. It has numereous applications in physics and applied mathematics. In this paper, we define an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This new transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. Furthermore, a discrete Weierstrass space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behaviour of the discrete delta functions and check if they are elements of this newly defined Weierstrass space.

General solution of the exceptional Hermite differential equation and its minimal surface representation

The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition $\lambda = (1)$ and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper-Weierstrass formula for the immersion in the Euclidean space $\mathbb{E}^3$, leading to minimal surfaces. Three-dimensional displays of these surfaces are presented.

The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew

We derive the formal expansion of the price of a VIX future (and more generally a VIX power payoff) in various Bergomi models at any order in powers of the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call implied volatility, and also expand this quantity at any order. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, the expansions are in closed form; otherwise, they involve one-dimensional integrals which are extremely fast to compute. Our extensive numerical experiments show that both the price and implied volatility expansions provide very accurate approximations for a wide range of model parameters which covers the values typically used in the equity and FX derivatives markets and even well beyond---in particular, the expansions are accurate even for very large values of the volatility-of-volatility. They also show that the implied volatility expansion converges much faster than the price expansion. Its truncation at order 1 is already so accurate that it leads to a simple formula for the price of the VIX future that can virtually be considered exact. The derivation of the expansions naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties; it is interesting in itself. The expansions allow us to precisely pinpoint the roles of all the model parameters (volatility-of-volatility, mean reversions, correlations, mixing fraction) in the formation of the prices of VIX futures and VIX power payoffs in Bergomi models. We also use them together with the Bergomi-Guyon expansion of the S&P 500 smile to (instantaneously) calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX2 implied volatility. Our tests and the new expansions shed more light on the inability of traditional stochastic volatility models to jointly fit S&P 500 and VIX market data. The (imperfect but decent) joint fit requires much larger values of volatility-of-volatility and fast mean reversion than the ones previously reported in Bergomi (2005, 2016).

Multiple Hermite polynomials and simultaneous Gaussian quadrature

Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to $r>1$ normal (Gaussian) weights $w_j(x)=e^{-x^2+c_jx}$ with different means $c_j/2$, $1 \\leq j \\leq r$. These polynomials have a number of properties such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated, and an interesting new feature is observed: depending on the distance between the $c_j$, $1 \\leq j \\leq r$, the zeros may accumulate on $s$ disjoint intervals, where $1 \\leq s \\leq r$. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form $\\displaystyle \\int_{-\\infty}^{\\infty} f(x) \\exp(-x^2 + c_jx)\\, dx$ simultaneously for $1 \\leq j \\leq r$ for the case $r=3$ and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights.

Electrostatic Problems with a Rational Constraint and Degenerate Lam\xe9 Equations

In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lamé differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain type of rational or polynomial constraint and polynomial solutions of a corresponding degenerate Lamé equation, see details below. In particular, the standard linear differential equations satisfied by the classical Hermite and Laguerre polynomials belong to this class. Besides these two classical cases, we present a number of other examples including some relativistic orthogonal polynomials and linear differential equations satisfied by those.

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero–Moser–Sutherland type and are related with β-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities k tend to ∞, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form k⋅x for k→∞ with x in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.

Electrostatic interpretation of zeros of orthogonal polynomials

We study the differential equation − ( p ( x ) y ′ ) ′ + q ( x ) y ′ = λ y , - (p(x) y’)’ + q(x) y’ = \lambda y, where p ( x ) p(x) is a polynomial of degree at most 2 and q ( x ) q(x) is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials, and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: a set of distinct, real numbers { x 1 , … , x n } \left \{x_1, \dots , x_n\right \} satisfies p ( x i ) ∑ k = 1 k ≠ i n 2 x k − x i = q ( x i ) − p ′ ( x i ) f o r a l l 1 ≤ i ≤ n \begin{equation*} p(x_i) \sum _{k = 1 \atop k \neq i}^{n}{\frac {2}{x_k - x_i}} = q(x_i) - p’(x_i) \qquad \mathrm {for all}~ 1\leq i \leq n \end{equation*} if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on p ( x ) , q ( x ) p(x),q(x) whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.

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).

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

Complex Exceptional Orthogonal Polynomials and Quasi-invariance

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1, \dots, \lambda_n)$ is a partition. G\'omez-Ullate et al showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of $\mathbb C[x]$ satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schr\"odinger operators, is also presented.

The spectral analysis of three families of exceptional Laguerre polynomials

The Bochner Classification Theorem (1929) characterizes the polynomial sequences {pn}n=0∞, with degpn=n that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences {pn}n=1∞, degpn=n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptionalX1-Laguerre and X1-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials {pn}n∈N0⧵A, where A is a finite subset of the non-negative integers N0 and where degpn=n for all n∈N0⧵A. We call such a sequence an exceptional polynomial sequence of codimension |A|, where the latter denotes the cardinality of A. All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting m polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as a complete spectral study of the second-order Type III exceptional Laguerre differential expression.

Complex Hermite functions as Fourier–Wigner transform

ABSTRACTWe prove that the complex Hermite polynomials on the complex plane can be realized as the Fourier–Wigner transform of the well-known real Hermite functions on the real line . This reduces considerably Wong's proof [Wong MW. Weyl transforms. New York: Universitext. Springer-Verlag; 1998. Chapter 21] giving the explicit expression of in terms of the Laguerre polynomials. Moreover, we derive some new integral identities for the classical real Hermite polynomials .

Accurate spectral numerical schemes for kinetic equations with energy diffusion

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker–Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

A NOTE ON A SPECIAL CLASS OF HERMITE POLYNOMIALS

This paper is devoted to the description of a special class of Hermite polynomials of five variables. It can be seen as an extension of the generalized vectorial Hermite polynomials of type Hm,n(x,y) and at the same time as a generalization of the Gould-Hopper Hermite polynomials of type Hn(x,y). We use the five-variable Hermite polynomials to derive reformulations of the well known operational relations satisfied from the generalized Hermite polynomials of different types.

Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux–Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2 m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2ℓ + 3 recurrence relation where ℓ is the length of the partition λ. Explicit expressions for such recurrence relations are given.

On Fourier integral transforms for q-Fibonacci and q-Lucas polynomials

We study in detail two families of q-Fibonacci polynomials and q-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and were then employed by Cigler and Zeng to construct novel q-extensions of classical Hermite polynomials. We show that both of these q-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform.