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The Real-rootedness of Eulerian Polynomials via the Hermite--Biehler Theorem  [PDF]
Arthur L. B. Yang,Philip B. Zhang
Mathematics , 2015,
Abstract: Based on the Hermite--Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $\mathbf{s}$-Eulerian polynomials. We also confirm Hyatt's conjectures on the interlacing property of half Eulerian polynomials. Borcea and Br\"and\'en's work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.
On integrals involving Hermite polynomials  [PDF]
D. Babusci,G. Dattoli,M. Quattromini
Mathematics , 2011,
Abstract: We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of gaussian functions and of multiple products of Hermite polynomials.
Generalized Hermite polynomials  [PDF]
Vadim V. Borzov
Mathematics , 2001,
Abstract: The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these polynomials satisfy the differential equation of the second order obtained in this work or there is no differential equation of a finite order for these polynomials.
Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials  [PDF]
Yan Xu
Mathematics , 2015,
Abstract: In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from the theorems directly. The relationship between Bernoulli and Euler polynomials are shown as a special case of the characterization theorem of the Appell sequence generated by $\alpha$ scaling functions.
Deformed Complex Hermite Polynomials  [PDF]
S. Twareque Ali,Mourad E. H. Ismail,Nurisya M. Shah
Physics , 2014,
Abstract: We study a class of bivariate deformed Hermite polynomials and some of their properties using classical analytic techniques and the Wigner map. We also prove the positivity of certain determinants formed by the deformed polynomials. Along the way we also work out some additional properties of the (undeformed) complex Hermite polynomials and their relationships to the standard Hermite polynomials (of a single real variable).
A deformation of Hermite polynomials  [PDF]
M. Mekhfi
Physics , 2000,
Abstract: We propose and study the properties of a set of polynomials $M_{n\alpha, H\ }^{s}(z)$, $C_{n\alpha, H}^{s}(z)$ $W_{n\alpha, H}^{s}(z)$ with $n,s\in N$ $;\alpha =\pm 1;$and where $H$ stands for Hermite ; the ''root '' polynomial >.These polynomials are obtained from a deformation of Hermite polynomials $H_{n}(z).$The structure underlying the deformation seems quite general and not only restricted to Hermite polynomials.
Generalized Hermite polynomials and the Bose-like oscillator calculus  [PDF]
Marvin Rosenblum
Mathematics , 1993,
Abstract: This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator calculus, which generalizes the calculus associated with the quantum mechanical simple harmonic oscillator, is studied in terms of these polynomials.
Zeros of exceptional Hermite polynomials  [PDF]
A. B. J. Kuijlaars,R. Milson
Physics , 2014,
Abstract: We study the zeros of exceptional Hermite polynomials associated with an even partition $\lambda$. We prove several conjectures regarding the asymptotic behavior of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor $\sqrt{2n}$, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial $H_{\lambda}$, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.
Hermite polynomials related to Genocchi, Euler and Bernstein polynomials  [PDF]
Serkan Araci,Jong Jin Seo,Mehmet Acikgoz
Mathematics , 2012,
Abstract: The objective of this paper is to derive some interesting properties of Genocchi, Euler and Bernstein polynomials by means of the orthogonality of Hermite polynomials.
On Hermite Matrix Polynomials of Two Variables  [PDF]
Ghazi S. Kahmmash
Journal of Applied Sciences , 2008,
Abstract: This study deals with the two-variable Hermite matrix polynomials, some relevant matrix functions appear interims of the two-variable Hermite matrix polynomials the relationships with Hermite matrix polynomials of one variable, Chepyshev matrix polynomials of the second kind have been obtained and expansion of the. Gegenbauer matrix polynomials as series of Hermite matrix polynomials.
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