A note on two-variable Chebyshev polynomials

Clemente Cesarano; Claudio Fornaro

doi:10.1515/gmj-2016-0034

A note on two-variable Chebyshev polynomials

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0034 ◽

2017 ◽

Vol 24 (3) ◽

pp. 339-349 ◽

Cited By ~ 9

Author(s):

Clemente Cesarano ◽

Claudio Fornaro

Keyword(s):

Chebyshev Polynomials ◽

Hermite Polynomials ◽

Integral Representations

AbstractIn this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.

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On a Class of Humbert-Hermite Polynomials

10.20944/preprints201909.0068.v1 ◽

2019 ◽

Author(s):

M. A Pathan ◽

Waseem Khan

Keyword(s):

Chebyshev Polynomials ◽

Hermite Polynomials ◽

Ultraspherical Polynomials

A unified presentation of a class of Humbert’s polynomials in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinnsy, Horadam-Pethe, Djordjević , Gould, Milovanović and Djordjević, Pathan and Khan polynomials and many not so called ’named’ polynomials has inspired the present paper and the authors define here generalized Humbert-Hermite polynomials of two variables. Several expansions of Humbert- Hermite polynomials, Hermite-Gegenbaurer (or ultraspherical) polynomials and Hermite- Chebyshev polynomials are proved.

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Defining relations of quantum symmetric pair coideal subalgebras

Forum of Mathematics Sigma ◽

10.1017/fms.2021.61 ◽

2021 ◽

Vol 9 ◽

Author(s):

Stefan Kolb ◽

Milen Yakimov

Keyword(s):

Chebyshev Polynomials ◽

Hermite Polynomials ◽

Symmetric Pair ◽

Star Products ◽

Graded Algebras ◽

Enveloping Algebras ◽

Defining Relations ◽

New Family

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$ -graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

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Integral representations and new generating functions of Chebyshev polynomials

Hacettepe Journal of Mathematics and Statistics ◽

10.15672/hjms.20154610029 ◽

2015 ◽

Vol 6 (1112) ◽

Cited By ~ 5

Author(s):

Clemente Cesarano

Keyword(s):

Chebyshev Polynomials ◽

Generating Functions ◽

Integral Representations

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Integral and series representations of q-polynomials and functions: Part I

Analysis and Applications ◽

10.1142/s0219530517500129 ◽

2018 ◽

Vol 16 (02) ◽

pp. 209-281 ◽

Cited By ~ 3

Author(s):

Mourad E. H. Ismail ◽

Ruiming Zhang

Keyword(s):

Integral Representation ◽

Orthogonal Polynomials ◽

Mellin Transform ◽

Bessel Functions ◽

Hermite Polynomials ◽

Integral Representations ◽

Moment Problems ◽

Exponential Functions ◽

Series Representations ◽

Indeterminate Moment Problems

By applying an integral representation for [Formula: see text], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [Formula: see text]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [Formula: see text]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [Formula: see text]-Hermite and [Formula: see text]-Hermite polynomials, and the [Formula: see text]-exponential functions [Formula: see text], [Formula: see text] and [Formula: see text]. Their representations are in turn used to derive many new identities involving [Formula: see text]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.

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Multi-Dimensional Chebyshev Polynomials: A Non-Conventional Approach

Communications in Applied and Industrial Mathematics ◽

10.1515/caim-2019-0008 ◽

2019 ◽

Vol 10 (1) ◽

pp. 1-19 ◽

Cited By ~ 4

Author(s):

Clemente Cesarano

Keyword(s):

Chebyshev Polynomials ◽

Polynomial Interpolation ◽

Hermite Polynomials ◽

Integral Transforms ◽

Multi Index ◽

Special Cases ◽

The Laplace Transform ◽

Translation Operators ◽

Sturm Liouville ◽

Symbolic Approach

Abstract Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

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ON SUMMABLE, POSITIVE POISSON–MEHLER KERNELS BUILT OF AL-SALAM–CHIHARA AND RELATED POLYNOMIALS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025712500142 ◽

2012 ◽

Vol 15 (03) ◽

pp. 1250014 ◽

Cited By ~ 2

Author(s):

PAWEŁ J. SZABŁOWSKI

Keyword(s):

Infinite Series ◽

Chebyshev Polynomials ◽

Hermite Polynomials ◽

Special Technique ◽

Linear Combinations

Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so-called Al-Salam–Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so-called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting nice identities built of the ASC polynomials involving six variables like e.g., formula (3.6). These expansions lead to asymmetric, positive and summable kernels. The particular cases (referring to q = 1 and q = 0) lead to the kernels build of certain linear combinations of the ordinary Hermite and Chebyshev polynomials.

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On similarity solutions of the differential equation ψzzzz + ψx = 0

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100057200 ◽

1970 ◽

Vol 67 (1) ◽

pp. 163-171 ◽

Cited By ~ 9

Author(s):

A. E. Gill ◽

R. K. Smith

Keyword(s):

Differential Equation ◽

Hermite Polynomials ◽

Fluid Flows ◽

Similarity Solutions ◽

Integral Representations ◽

Parabolic Cylinder ◽

Coriolis Parameter ◽

Similarity Functions ◽

Image Position ◽

Boussinesq Fluid

AbstractSolutions of the partial differential equation ψzzzz + ψx + 0 of the type ψ = x¼nw(y), where y = zx−¼ and n is an integer, are investigated. The equation occurs as a boundary-layer approximation in certain rotating and stratified fluid flows in which the production of vorticity (due, for example, to changes in the Coriolis parameter with latitude in two-dimensional flows on a beta plane, or by the buoyant generation of vorticity in a Boussinesq fluid) is opposed by a diffusive process. The similarity functions w(y) satisfy the fourth order differential equation.These functions have many properties analogous to those of error functions and parabolic cylinder functions. When n is a non-negative integer, there exist polynomial solutions of the latter equation. These have analogies with Hermite polynomials, although they do not form an orthogonal set in any useful sense. The main properties of the similarity functions are listed, including their series expansions, integral representations, asymptotic expansions and recurrence relations. In particular, a pair of independent solutions, Jon(y) and Kon(y), are denned such that Jon(y) and Kon(y) are real for real y and vanish at an exponential rate as y → + ∞. The function Jon(y) also decays exponentially as y → − ∞ if n is negative, and grows algebraically as y → − ∞ if n is positive. Curves of the functions Jon(y) and Kon(y) are given for |n| ≤ 3 and 0 < y < 6, and their more useful properties are listed.

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