Search Results (57)

This paper establishes a new characterization of symmetric q-Dunkl-classical orthogonal polynomials through a second structure relation. These symmetric polynomials generalize the $q^2$-analogues of Hermite and Gegenbauer polynomials. Our main result provides a finite expansion of each polynomial in terms of its q-Dunkl derivatives, offering a new effective classification method. We derive explicit structure relations for the $q^2$-analogue of generalized Hermite and the $q^2$-analogue of generalized Gegenbauer polynomials. Full article

This study presents the Fourier–Gegenbauer integral Galerkin (FGIG) method, a new numerical framework that uniquely integrates Fourier series and Gegenbauer polynomials to solve the one-dimensional advection–diffusion (AD) equation with spatially symmetric periodic boundary conditions, achieving exponential convergence and reduced computational cost compared to traditional methods. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Unlike traditional approaches, this method eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of the shifted Gegenbauer–Gauss (SGG) quadrature to ensure high accuracy and stability for relatively low Péclet numbers. This approach simplifies calculations of integrals, making the method faster and more reliable for diverse problems. Numerical experiments presented validate the method’s superior performance over traditional techniques, such as finite difference, finite element, and spline-based methods, achieving near-machine precision with significantly fewer mesh points. These results demonstrate its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering. The method’s fusion of spectral precision and integral reformulation marks a significant advancement in numerical PDE solvers, offering a scalable, high-fidelity alternative to conventional time-stepping techniques. Full article

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation matrix (SGDM), while other terms are computed with precise quadrature rules. By integrating advanced techniques such as the shifted Gegenbauer pseudospectral method (SGPS), fractional derivative and integral approximations, and barycentric integration matrices, the HSG-IPS method achieves spectral accuracy. Numerical results show it reduces average absolute errors (AAEs) by up to 99.99% compared to methods like Crank–Nicolson linearized difference scheme (CNLDS) and finite integration method using Chebyshev polynomial (FIM-CBS), with computational times as low as 0.04–0.05 s. The method’s stability is improved by avoiding ill-conditioned high-order derivative approximations, and its efficiency is boosted by precomputed matrices and Kronecker product structures. Robust across various fractional orders, the HSG-IPS method offers a powerful tool for modeling wave propagation and nonlinear phenomena in fractional calculus applications. Full article

This paper introduces a Gegenbauer-based fractional approximation (GBFA) method for high-precision approximation of the left Riemann–Liouville fractional integral (RLFI). By using precomputable fractional-order shifted Gegenbauer integration matrices (FSGIMs), the method achieves super-exponential convergence for smooth functions, delivering near machine-precision accuracy with minimal computational cost. Tunable shifted Gegenbauer (SG) parameters enable flexible optimization across diverse problems, while rigorous error analysis confirms rapid error decay under optimal settings. Numerical experiments demonstrate that the GBFA method outperforms MATLAB’s integral, MATHEMATICA’s NIntegrate, and existing techniques by up to two orders of magnitude in accuracy, with superior efficiency for varying fractional orders $0<\alpha <1$. Its adaptability and precision make the GBFA method a transformative tool for fractional calculus, ideal for modeling complex systems with memory and non-local behavior. Full article

This paper introduces a novel Shifted Gegenbauer Pseudospectral (SGPS) method for approximating Caputo fractional derivatives (FDs) of an arbitrary positive order. The method employs a strategic variable transformation to express the Caputo FD as a scaled integral of the mth-derivative of the Lagrange interpolating polynomial, thereby mitigating singularities and improving numerical stability. Key innovations include the use of shifted Gegenbauer (SG) polynomials to link mth-derivatives with lower-degree polynomials for precise integration via SG quadratures. The developed fractional SG integration matrix (FSGIM) enables efficient, pre-computable Caputo FD computations through matrix–vector multiplications. Unlike Chebyshev or wavelet-based approaches, the SGPS method offers tunable clustering and employs SG quadratures in barycentric forms for optimal accuracy. It also demonstrates exponential convergence, achieving superior accuracy in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley–Torvik type. The method unifies interpolation and integration within a single SG polynomial framework and is extensible to multidimensional fractional problems. Full article

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 integral formula that expresses the Jacobi polynomials in terms of Gegenbauer polynomials, indicating a general procedure to extend Askey’s scheme of classical polynomials by one step. We can also formulate suitably normalized Fourier–Jacobi spectral coefficients of a function in terms of the Fourier cosine coefficients of a proper Abel-type transform involving a fractional integral of the function itself. This new means of representing the spectral coefficients can be beneficial for the numerical analysis of fractional differential and variational problems. Moreover, the symmetry properties made explicit by this representation lead us to identify the classes of Jacobi polynomials that naturally admit the extension of the definition to negative values of the index. Examples of the application of this representation, aiming to prove the properties of the Fourier–Jacobi spectral coefficients, are finally given. Full article

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 function of the first kind $J_N\left(kx\right)$ and modified Bessel functions of the first kind $I_N\left(kx\right)$ lead to an infinite set of series involving ${}_{1}F_2$ hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving ${}_{1}F_2$ hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving ${}_{1}F_2$ hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers’ equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of solving the differential problem together with the conditions, we solve a system of algebraic equations. The method can handle complex boundary conditions. Some numerical experiments are exhibited to demonstrate that this approach is highly efficient and produces results that are better than some existing numerical methods in the literature. This technique offers more advanced solutions for time-fractional problems in various fields. Full article

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields. Full article

The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials $C_n^{\left(\nu \right)}\left(x\right)$, where $\nu >-{\displaystyle \frac{1}{2}}$. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as ${}_x\mathcal{D}^{\alpha }$, providing an alternative framework for fractional calculus. Full article

This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families. Full article

Analyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier-series expansion of the function ${(1-2\alpha cos\theta +{\alpha }^2)}^{-s}$, we obtain a family of new expansions and generating functions for the Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials $T_n$ and the Gegenbauer polynomials $C_n^{\left(2\right)}$ is given. Full article

The purpose of this article is to introduce and study certain families of normalized certain functions with symmetric points connected to Gegenbauer polynomials. Moreover, we determine the upper bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ and resolve the Fekete–Szegöproblem for these functions. In addition, we establish links to a few of the earlier discovered outcomes. Full article

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulfill either Hanh or Appell conditions. Full article

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated. Full article