Results 251 to 260 of about 77,211 (287)

Unsupervised meta-analysis on chemical elements and atomic energy prediction: A case study on the periodic table. [PDF]

Heliyon
Belahcene B.
europepmc   +1 more source

Variational calculus approach to Zernike polynomials with application to fluorescence correlation spectroscopy

Gligonov I, Enderlein J.
europepmc   +1 more source

MDS, Hermitian almost MDS, and Gilbert-Varshamov quantum codes from generalized monomial-Cartesian codes. [PDF]

Quantum Inf Process
Barbero-Lucas B, Hernando F, Martín-Cruz H, McGuire G.   +3 more
europepmc   +1 more source

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Classical Orthogonal Polynomials

1999
Example 7.2.5 discussed only one of the many types of the so-called classical orthogonal polynomials. Historically, these polynomials were discovered as solutions to differential equations arising in various physical problems. Such polynomials can be produced by starting with 1,x,x 2,… and employing the Gram-Schmidt process.
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On bivariate classical orthogonal polynomials

Applied Mathematics and Computation, 2018
Abstract We deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi-definite moment functional, and we revise old properties for these polynomials. More precisely, new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal Pearson-type equation are proved: they are solutions of
Teresa E. Pérez, Miguel A. Piñar, Francisco Marcellán, Francisco Marcellán, Misael E. Marriaga   +4 more
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Classical Continuous Orthogonal Polynomials [PDF]

, 2020
Classical orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel) constitute the most important families of orthogonal polynomials. They appear in mathematical physics when Sturn-Liouville problems for hypergeometric differential equation are studied. These families of orthogonal polynomials have specific properties. Our main aim is to: 1.
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Another Characterization of the Classical Orthogonal Polynomials

SIAM Journal on Mathematical Analysis, 1972
The classical orthogonal polynomials of Jacobi, Laguerre and Hermite are characterized as the only orthogonal polynomials with a differentiation formula of the form \[ \pi (x)P'_n (x) = \left( {\alpha _n x + \beta _n } \right)P_n (x) + \gamma _n P_{n - 1} (x),\quad n \geqq 1,\] where $\pi (x)$ is a polynomial.
T. S. Chihara, W. A. Al-Salam
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Classical orthogonal polynomials: A functional approach

Acta Applicandae Mathematicae, 1994
The Rodriguez formula for the classical orthogonal polynomials (here Hermite, Laguerre, Jacobi and Bessel) leads to a distributional differential equation for the corresponding moment functional \(u\) of the form \(D(\Phi u)=\Psi u\) with polynomials \(\Phi,\Psi\) and \(\deg \Phi \leq 2\), \(\deg \Psi=1\).
J. Petronilho, Francisco Marcellán, Amílcar Branquinho   +2 more
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Discrete classical orthogonal polynomials

Journal of Difference Equations and Applications, 1998
We find necessary and sufficient conditions for the difference equation of hypergeometric type to have polynomial solutions , which are orthogonal, that is Traditionallydμ(x) is a positive measure but here we allow it to be a signed measure. We then show that the usual restrictions on parameters in discrete classical orthogonal polynomials can be ...
Kwon, KH Kwon, Kil Hyun, Lee, DW, Park, SB, Yoo, BH   +3 more
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Positive Sums of the Classical Orthogonal Polynomials

SIAM Journal on Mathematical Analysis, 1977
An expansion as a sum of squares of Jacobi polynomials \(P_n^{(\alpha , \beta )}(x)\) is used to prove that if \(0 \leq \lambda \leq \alpha + \beta\) and \(\beta \geq -1/2\), then \[ \sum_{k=1}^{n} \frac{(\lambda +1)_{n-k}}{(n-k)!} \frac{(\lambda +1)_k}{k!} \frac{P_k^{(\alpha ,\beta )}(x)}{P_k^{(\alpha ,\beta )}} = 0,\quad -1\leq x \leq \infty, \tag{\(*
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