Zeros of the Exceptional Laguerre and Jacobi Polynomials

Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽

10.3390/sym10110617 ◽

2018 ◽

Vol 10 (11) ◽

pp. 617 ◽

Cited By ~ 5

Author(s):

Dmitry Dolgy ◽

Dae Kim ◽

Taekyun Kim ◽

Jongkyum Kwon

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Jacobi Polynomials ◽

Linear Combinations ◽

The Third ◽

Connection Problem ◽

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

10.20944/preprints201810.0140.v1 ◽

2018 ◽

Author(s):

Dmitry Victorovich Dolgy ◽

Dae San Kim ◽

Taekyun Kim ◽

Jongkyum Kwon

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Jacobi Polynomials ◽

Linear Combinations ◽

The Third ◽

Connection Problem ◽

Expressing Sums of Finite Products of Chebyshev Polynomials of the Second Kind and of Fibonacci Polynomials by Several Orthogonal Polynomials

Mathematics ◽

10.3390/math6100210 ◽

2018 ◽

Vol 6 (10) ◽

pp. 210 ◽

Cited By ~ 13

Author(s):

Taekyun Kim ◽

Dae Kim ◽

Jongkyum Kwon ◽

Dmitry Dolgy

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Jacobi Polynomials ◽

Linear Combinations ◽

Fibonacci Polynomials ◽

This paper is concerned with representing sums of the finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations, each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, which involve the hypergeometric functions 1 F 1 and 2 F 1 .

Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane

Constructive Approximation ◽

10.1007/s00365-020-09515-0 ◽

2020 ◽

Author(s):

Gernot Akemann ◽

Taro Nagao ◽

Iván Parra ◽

Graziano Vernizzi

Keyword(s):

Orthogonal Polynomials ◽

Complex Plane ◽

Jacobi Polynomials ◽

Complete Metric Space ◽

Hermite Polynomials ◽

Weighted Bergman Space ◽

Space Forms ◽

Entire Complex ◽

Fourth Kind

Abstract We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $$C_n^{(1+\alpha )}(z)$$ C n ( 1 + α ) ( z ) for $$\alpha >-1$$ α > - 1 containing the Legendre polynomials $$P_n(z)$$ P n ( z ) and the subset $$P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)$$ P n ( α + 1 2 , ± 1 2 ) ( z ) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit $$\alpha \rightarrow \infty $$ α → ∞ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.

Expressing Sums of Finite Products of Chebyshev Polynomials of the Second Kind and of Fibonacci Polynomials by Several Orthogonal Polynomials

10.20944/preprints201809.0258.v1 ◽

2018 ◽

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Jongkyum Kwon ◽

Dmitry V. Dolgy

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Jacobi Polynomials ◽

Linear Combinations ◽

Fibonacci Polynomials ◽

This paper is concerned with representing sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials which involve the hypergeometric functions ${}_1 F_1$ and ${}_2 F_1$.

On the Galois group of Generalised Laguerre polynomials II

10.46298/hrj.2020.6457 ◽

2020 ◽

Vol Volume 42 - Special... ◽

Author(s):

Shanta Laishram ◽

Saranya G. Nair ◽

T. N. Shorey

Keyword(s):

Numerical Analysis ◽

Orthogonal Polynomials ◽

Real Number ◽

Galois Group ◽

Laguerre Polynomials ◽

Mathematical Physics ◽

Negative Integer ◽

Algebraic Properties ◽

International Audience

International audience For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by$$L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}.$$These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ when $u$ is a negative integer.

Orthogonal polynomials satisfying fourth order differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015213 ◽

1981 ◽

Vol 87 (3-4) ◽

pp. 271-288 ◽

Cited By ~ 80

Author(s):

Allan M. Krall

Keyword(s):

Differential Equation ◽

Orthogonal Polynomials ◽

Fourth Order ◽

Order Differential Equation ◽

Jacobi Polynomials ◽

Weight Functions ◽

Fourth Order Differential Equation

Absolutely Continuous ◽

Independent Variable ◽

SynopsisThese polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.

Potential energies of one-dimensional systems at equilibrium

Canadian Journal of Physics ◽

10.1139/p81-110 ◽

1981 ◽

Vol 59 (7) ◽

pp. 859-862

Author(s):

Shafique Ahmed

Keyword(s):

Dynamical Systems ◽

Orthogonal Polynomials ◽

Jacobi Polynomials ◽

Bessel Polynomials ◽

One Dimensional ◽

The One

The equilibrium positions of some one-dimensional systems coincide with the zeros of Hermite, Laguerre, and Jacobi polynomials. Using the discriminants of these classical orthogonal polynomials it is then possible to calculate the values of the potential energies of the one-dimensional systems at equilibrium. It is also shown that the zeros of Bessel polynomials coincide with the equilibrium positions of certain dynamical systems.