In this paper, we present a new integral representation for the Jacobi polynomials that follows from Koornwinder’s representation by introducing a suitable new form of Euler’s formula. From this representation, we obtain a fractional inte...
In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete–continuous Sobolev-type inner product defined in terms of the Jacobi measure.
In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable...
The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential po...
The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (GJPs). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of...
The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation...
Using the Sturm–Liouville theory in q-difference spaces, we prove the finite orthogonality of q-Pseudo Jacobi polynomials. Their norm square values are then explicitly computed by means of the Favard theorem.
Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-cal...
We study the sequence of monic polynomials {Sn}n⩾0, orthogonal with respect to the Jacobi-Sobolev inner product ⟨f,g⟩s=∫−11f(x)g(x)dμα,β(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where N,dj∈Z...
This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equat...
This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new f...
The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we...
In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential opera...
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 key expressi...
The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients wh...
The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as...
This study explores the application of Romanovski–Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions that meet the homogeneous ini...
Within eye diseases, diabetic retinopathy and retinopathy of prematurity are considered one of the main causes of blindness in adults and children. In order to prevent the disease from reaching such an extreme, a timely diagnosis and effective treatm...
Numerous researchers have extensively studied various subfamilies of the bi-univalent function family utilizing special functions. In this paper, we introduce and investigate a new subfamily of bi-univalent functions, which is defined on the symmetri...
In the present paper, we numerically simulate fractional-order model of the Bloch equation by using the Jacobi polynomials. It arises in chemistry, physics and nuclear magnetic resonance (NMR). It also arises in magnetic resonance imaging (MRI) and e...
Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The...
This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for sol...
In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshe...
The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solva...
The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim t...
In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to comple...
The lightweight of structure is widely applied in industrial applications, and the conflict between both dynamic stability and structural lightweight is still prominent. In this paper, functionally graded porous (FGP) elliptic cylindrical shells and...
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 e...
Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed...
We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fracti...
We obtain an analytic approximation of the bound states solution of the Schrödinger equation on the semi-infinite real line for two potential models with a rich structure as shown by their spectral phase diagrams. These potentials do not belong...
We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operation...
In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. W...
In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In add...
In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m&is...
A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM)...
A general formulation is considered for the free vibration of curved laminated composite beams (CLCBs) with alterable curvatures and diverse boundary restraints. In accordance with higher-order shear deformation theory (HSDT), an improved variational...
In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs...
This paper seeks to establish a generalized numerical model to examine the free vibration behavior of functionally graded porous (FGP) elliptical shells and panels with various boundary types. The model is built on first-order shear deformation theor...
The design of a hypersonic vehicle controller has been an active research field in the last decade, especially when the vehicle is studied as a time-varying system. A time-varying compound control method is proposed for a hypersonic vehicle controlle...
Electromagnetic actuator systems composed of an induction servo motor (ISM) drive system and a rice milling machine system have widely been used in agricultural applications. In order to achieve a finer control performance, a witty control system usi...
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researche...
With the flourishing of social media platforms, data in social networks, especially user-generated content, are growing rapidly, which makes it hard for users to select relevant content from the overloaded data. Recommender systems are thus developed...
The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f...
Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-e...
In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcenden...
The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These G...
This paper presents a novel numerical approach to handling ordinary differential equations (ODEs) with initial conditions (ICs) by introducing generalized exponential Jacobi functions (GEJFs). These GFJFs satisfy the associated ICs. A crucial part of...
In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyze the walks on the glued trees, we consider a reduction to the walks on path graphs. Using a spectral analysis of the Jacob...
Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel functi...
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