On Hermite expansion of $x^p_+$

Zbigniew Sadlok

Displaying similar documents to “On Hermite expansion of $x_{+}^{p}$ ”

Note on the multidimensional Gebelein inequality

Marek Beśka, Agnieszka Wałachowska (2013)

Applicationes Mathematicae

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We generalize the Gebelein inequality for Gaussian random vectors in $ℝ^{d}$ .

On Hermite expansions of $x^{k}$ and ln|x|

Z. Sadlok, Z. Tyc (1977)

Annales Polonici Mathematici

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Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. II

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. I

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

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Let $P_{k}$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_{k}$ in $f = \sum_{k} a_{k} P_{k}$ . A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_{k}$ results in obtaining a low order of the recurrence.

Deformed Heisenberg algebra with reflection and $d$ -orthogonal polynomials

Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani (2017)

Czechoslovak Mathematical Journal

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This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of $d$ -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when $d = 1$ . The underlying algebraic framework allowed a systematic...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of $ℂ^{N}$ . We define a pluricomplex Green’s function $V_{K, θ}$ for θ-incomplete polynomials. We establish properties of $V_{K, θ}$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of $ℝ^{N}$ , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K ∖ s u p p {(d d^{c} V_{K, θ})}^{N}$ . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $s u p p {(d d^{c} V_{K, θ})}^{N}$ when K is a compact...

A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera, José A. Díaz-García, Graciela González-Farías (2007)

Applicationes Mathematicae

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This work solves the partial differential equation for Jack polynomials $C_{κ}^{α}$ of the second order. When the parameter α of the solution takes the values 1/2, 1 and 2 we get explicit formulas for the quaternionic, complex and real zonal polynomials of the second order, respectively.

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Remarks on q-CCR relations for |q| > 1

Marek Bożejko (2007)

Banach Center Publications

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In this paper we give a construction of operators satisfying q-CCR relations for q > 1: $A (f) A * (g) - A * (g) A (f) = q^{N} ⟨ f, g ⟩ I$ and also q-CAR relations for q < -1: $B (f) B * (g) + B * (g) B (f) = {| q |}^{N} ⟨ f, g ⟩ I$ , where N is the number operator on a suitable Fock space $ℱ_{q} ()$ acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $∥ \cdot ∥$ be the uniform norm in the unit disk. We study the quantities $M_{n} (α) : = inf (∥ z P (z) + α ∥ - α)$ where the infimum is taken over all polynomials $P$ of degree $n - 1$ with $∥ P (z) ∥ = 1$ and $α > 0$ . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{α > 0} M_{n} (α) = 1 / n$ . We find the exact values of $M_{n} (α)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

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We find examples of polynomials $f \in D [t; σ, δ]$ whose eigenring $ℰ (f)$ is a central simple algebra over the field $F = C \cap Fix (σ) \cap Const (δ)$ .

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

Jeffrey S. Geronimo, Plamen Iliev (2014)

Journal of the European Mathematical Society

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We give a complete characterization of the positive trigonometric polynomials $Q (θ, ϕ)$ on the bi-circle, which can be factored as $Q (θ, ϕ) = | p (e^{i θ}, e^{i ϕ}) |^{2}$ where $p (z, w)$ is a polynomial nonzero for $| z | = 1$ and $| w | \leq 1$ . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4 π^{2} Q (θ, ϕ)}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of $L_{p}$ functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative $L_{p}$ spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given ${h_{1}, \dots, h_{t}}$ a finite subset of $ℝ^{d}$ , we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $ℝ^{d}$ with the property that the forward differences $Δ_{h_{k}}^{m_{k}} f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_{1}, \dots, m_{t}$ .

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ in the special case where $f (x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f (x)$ is monic and linear.

Thom polynomials and Schur functions: the singularities $I I I_{2, 3} (-)$

Özer Öztürk (2010)

Annales Polonici Mathematici

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We give a closed formula for the Thom polynomials of the singularities $I I I_{2, 3} (-)$ in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.

Approximation by weighted polynomials in $ℝ^{k}$

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in $ℝ^{k}$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $Σ \subset ℝ^{k}$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

Displaying similar documents to “On Hermite expansion of x + p ”

Displaying similar documents to “On Hermite expansion of $x_{+}^{p}$ ”