Jacobi polynomials
View article: CONSTRUCTING A TWO-VARIABLE ANALOGUE OF EXTENDED JACOBI POLYNOMIALS
CONSTRUCTING A TWO-VARIABLE ANALOGUE OF EXTENDED JACOBI POLYNOMIALS Open
The two-variable analogue of extended Jacobi polynomials has been researched. Orthogonality and differential operator connections describe these twovariable Jacobi polynomials, which generalize the classical Jacobi polynomials. A coupled p…
View article: CONSTRUCTING A TWO-VARIABLE ANALOGUE OF EXTENDED JACOBI POLYNOMIALS
CONSTRUCTING A TWO-VARIABLE ANALOGUE OF EXTENDED JACOBI POLYNOMIALS Open
The two-variable analogue of extended Jacobi polynomials has been researched. Orthogonality and differential operator connections describe these twovariable Jacobi polynomials, which generalize the classical Jacobi polynomials. A coupled p…
View article: Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme Open
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics w…
View article: On semi-separability and differentiation matrices
On semi-separability and differentiation matrices Open
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
View article: Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme Open
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics w…
View article: Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications
Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications Open
The aim of this study is to show that harmonic geometric polynomials can be represented in terms of geometric polynomials. This problem was first considered by Keller [14]; however, the corresponding coefficients were not fully determined.…
View article: Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications
Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications Open
The aim of this study is to show that harmonic geometric polynomials can be represented in terms of geometric polynomials. This problem was first considered by Keller [14]; however, the corresponding coefficients were not fully determined.…
View article: On semi-separability and differentiation matrices
On semi-separability and differentiation matrices Open
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
View article: Polynomials of the Askey scheme as Clebsch-Gordan coefficients
Polynomials of the Askey scheme as Clebsch-Gordan coefficients Open
Given a semi-simple algebra equipped with a coproduct, the Clebsch--Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-…
View article: Robust Approximation for Non-Linear Variable-Distributed Fractional Differential Equation with Non-Smooth Solutions
Robust Approximation for Non-Linear Variable-Distributed Fractional Differential Equation with Non-Smooth Solutions Open
This article introduces a spectral method aimed at estimating solutions for nonlinear Variable Distributed-Order Fractional Differential Equations (VDO-FDEs) with a non-smooth solution in one-dimensional and time-nonlinear VDOFDEs. Initial…
View article: Comparative Study of Shifted Chebyshev Polynomials on the Solution of Nonlinear Boundary Value Problems
Comparative Study of Shifted Chebyshev Polynomials on the Solution of Nonlinear Boundary Value Problems Open
The usefulness of orthogonal polynomials has increasingly been extended to the solution of initial and boundary value problems in recent years. Among these, Chebyshev polynomials—classified into four distinct kinds—are widely employed; how…
View article: Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats
Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats Open
Let $M$ be a product of rank-one symmetric spaces of compact type, each of dimension at least $3$. We establish sharp $L^p$ bounds for the restriction of Laplace--Beltrami eigenfunctions on $M$ to arbitrary submanifolds contained in a maxi…
View article: Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats
Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats Open
Let $M$ be a product of rank-one symmetric spaces of compact type, each of dimension at least $3$. We establish sharp $L^p$ bounds for the restriction of Laplace--Beltrami eigenfunctions on $M$ to arbitrary submanifolds contained in a maxi…
View article: Orthogonal polynomials with respect to a q-difference operator existence and uniqueness
Orthogonal polynomials with respect to a q-difference operator existence and uniqueness Open
This study expands the concept of orthogonality to orthogonal polynomials with respect to a q -difference operator. We investigate the existence of polynomials orthogonal with respect to a q -difference operator, and provide three differen…
View article: Christoffel functions, $$L^p$$ Markov inequalities and Marcinkiewicz-Zygmund type discretization for homogeneous polynomials on convex bodies
Christoffel functions, $$L^p$$ Markov inequalities and Marcinkiewicz-Zygmund type discretization for homogeneous polynomials on convex bodies Open
In this paper we will study various extremal problems related to multivariate homogeneous polynomials on convex bodies. We will construct certain homogeneous needle polynomials on convex bodies which will play a central role in the study o…
View article: Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics
Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics Open
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,α) = {x^α}{\mathrm e^{- x-\frac{t}{x}}}, \; …
View article: Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics
Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics Open
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,α) = {x^α}{\mathrm e^{- x-\frac{t}{x}}}, \; …
View article: Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance
Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance Open
In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.0331…
View article: Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance
Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance Open
In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.0331…
View article: Spectral algorithm for fractional BVPs via novel modified Chebyshev polynomials fractional derivatives
Spectral algorithm for fractional BVPs via novel modified Chebyshev polynomials fractional derivatives Open
This paper introduces a spectral algorithm tailored for solving fractional boundary value problems (BVPs) using the fractional derivatives of modified Chebyshev polynomials. Specifically, it addresses linear and non-linear BVPs and Bratu e…
View article: Arithmetic and geometry of Markov polynomials
Arithmetic and geometry of Markov polynomials Open
Markov polynomials are the Laurent-polynomial solutions of the generalised Markov equation X2 + Y 2 + Z2 = kXY Z, k = x2 + y2 + z2 xyz which are the results of cluster mutations applied to the initial triple (x, y, z). They were first in…
View article: The Imaginary Error Function and New Classes of Bi-Univalent Functions Subordinate to Jacobi Polynomials
The Imaginary Error Function and New Classes of Bi-Univalent Functions Subordinate to Jacobi Polynomials Open
A unique family of bi-univalent functions, commonly known as functions that are defined on the symmetric domain, is presented and investigated in this paper. We also presented and examined the subfamily of the functions. The imaginary erro…
View article: A Study of the Expansion of the Polynomial Set An {(Xm),y} in Terms of Jacobi Polynomials
A Study of the Expansion of the Polynomial Set An {(Xm),y} in Terms of Jacobi Polynomials Open
Here in this paper a systematic study of the expansion of the multivariable generalized polynomial set An{(xm), y} has been presented in a finite series consisting product of Jacobi polynomials in Lauricella form. A number of new useful re…
View article: A Study of the Expansion of the Polynomial Set An {(Xm),y} in Terms of Jacobi Polynomials
A Study of the Expansion of the Polynomial Set An {(Xm),y} in Terms of Jacobi Polynomials Open
Here in this paper a systematic study of the expansion of the multivariable generalized polynomial set An{(xm), y} has been presented in a finite series consisting product of Jacobi polynomials in Lauricella form. A number of new useful re…
View article: Novel Formulas of Specific Non-Symmetric Jacobi Polynomials with an Application in Numerical Analysis
Novel Formulas of Specific Non-Symmetric Jacobi Polynomials with an Application in Numerical Analysis Open
This paper introduces new formulas for non-symmetric Jacobi polynomials of specific parameters, focusing specifically on the subclasses where the difference between the two parameters of Jacobi polynomials is two or three. First, several k…
View article: CERTAIN FINITE INTEGRAL FORMULAS PERTAINING TO THE PRODUCT OF A GENERALIZED BESSEL-MAITLAND FUNCTION AND JACOBI POLYNOMIAL
CERTAIN FINITE INTEGRAL FORMULAS PERTAINING TO THE PRODUCT OF A GENERALIZED BESSEL-MAITLAND FUNCTION AND JACOBI POLYNOMIAL Open
View article: Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann–Hilbert perspective
Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann–Hilbert perspective Open
In this work we show how to get advantage from the Riemann–Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We deduc…
View article: Hypergeometric Functions as Activation Functions: The Particular Case of Bessel-Type Functions
Hypergeometric Functions as Activation Functions: The Particular Case of Bessel-Type Functions Open
The choice of the activation functions in neural networks (NN) are of paramount importance in the training process and the performance of NNs. Therefore, the machine learning community has directed its attention to the development of compu…
View article: Recurrence Relations for Jacobi Orthogonal Polynomials on the Triangular Domain
Recurrence Relations for Jacobi Orthogonal Polynomials on the Triangular Domain Open
In this paper, we present recurrence relations for the Jacobi weighted orthogonal polynomials Pn,r(α,β,γ) (u, v, w) with r = 0, 1, . . . , n, where n ≥ 0, defined on the triangular domain T = {(u, v, w) : u, v, w ≥ 0, u + v + w = 1} for va…
View article: Explicit representations of the norms of the Laguerre-Sobolev and Jacobi-Sobolev polynomials
Explicit representations of the norms of the Laguerre-Sobolev and Jacobi-Sobolev polynomials Open
This paper deals with discrete Sobolev orthogonal polynomials with respect to inner products built upon the classical Laguerre and Jacobi measures on the intervals $$ [0,\infty ) $$ and $$ [-1,1] $$ , respectively. In additi…