Search Results (45)

We study fractional and complex powers of a fixed directional derivative in ${\mathbb{R}}^d$, defined via a Marchaud-type singular integral representation. Under explicit convergence assumptions, this yields a pointwise nonlocal realization along rays. We then formulate a Ramanujan–Hardy approach to fractional directional differentiation based on analytic interpolation of the directional jet at a point. This construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. We emphasize that the resulting Ramanujan–Hardy derivative is defined through a Hardy-admissible interpolant of the directional jet. As an application, we investigate fractional directional derivatives of the Newtonian kernel in dimension $d\ge 3$. After a justified regularization and reduction to a Marchaud-type integral, we obtain a one-dimensional integral representation and a zonal harmonic description of the resulting function. This leads to a fractional Maxwell–Gegenbauer identity for $0<\Re \left(s\right)<1$, expressing the fractional directional derivative of ${\parallel x\parallel }^{2-d}$ in terms of Gegenbauer functions of complex degree. In this way, the classical Maxwell multipole formula appears as the integer-order case of a continuous analytic family. Moreover, the fractional operator preserves the main structural properties of the Newtonian kernel, including homogeneity, rotational invariance, and harmonicity away from the origin. The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere, while clarifying the distinction between Marchaud and jet-interpolation constructions of fractional directional operators. Full article

This paper presents modified Lagrange–Jacobi functions derived from the sine, exponential, and hyperbolic tangent coordinate transformations. The resulting Lagrange–Jacobi functions and their respective matrix elements for observables can be reduced to their respective Lagrange–Legendre, Lagrange–Chebyshev, and Lagrange–Gegenbauer functions. Furthermore, this paper postulates that the Lagrange-mesh functions form an approximate complete set of basis, a property implied by their approximate orthogonality. 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

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

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series. 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

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

Subclasses of analytic and bi-univalent functions have been extensively improved and utilized for estimating the Taylor–Maclaurin coefficients and the Fekete–Szegö functional. In this paper, we consider a certain subclass of normalized analytic and bi-univalent functions. These functions have inverses that possess a bi-univalent analytic continuation to an open unit disk and are associated with orthogonal polynomials; namely, Gegenbauer polynomials that satisfy subordination conditions on the open unit disk. We use this subclass to derive new approximations for the second and third Taylor–Maclaurin coefficients and the Fekete–Szegö functional. Furthermore, we discuss several new results that arise when we specialize the parameters used in our fundamental findings. Full article

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 solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes. Full article

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the q-derivative operator $D_q\left(0<q<1\right)$ and the Gegenbauer polynomials in a symmetric domain, which is the open unit disc $\mathsf{\Lambda }=\left\{\wp :\wp \in \mathbb{C}\mathrm{and}\left|\wp \right|<1\right\}.$ For these subclasses of analytic and bi-univalent functions, the coefficient estimates and Fekete–Szegö inequalities are solved. Some special cases of the main results are also linked to those in several previous studies. The symmetric nature of quantum calculus itself motivates our investigation of the applications of such quantum (or q-) extensions in this paper. Full article

In the real world, there are many applications that find the Pascal distribution to be a useful and relevant model. One of these is the normal distribution. In this work, we develop a new subclass of analytic bi-univalent functions by making use of the q-Pascal distribution series as a construction. These functions involve the q-Gegenbauer polynomials, and we use them to establish our new subclass. Moreover, we solve the Fekete–Szegö functional problem and analyze various different estimates of the Maclaurin coefficients for functions that belong to the new subclass. Full article

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by a fluid fictitious envelope. The fourth order partial differential equation governing the flow was completed with Happel-type boundary conditions which dictate no fluid slip on the inverted spheroid and a shear stress free non-permeable fictitious boundary. Through innovative means, such as the Kelvin inversion method and the R-semiseparation technique, a stream function was obtained as series expansion of Gegenbauer functions of the first and the second kinds of even order. Based on this, analytical expressions of meaningful hydrodynamic quantities, such as the velocity and the pressure field, were calculated and depicted in informative graphs. Using the first term of the stream function, the drag force exerted on the erythrocyte and the drag coefficient were calculated relative to the solid volume fraction of the cell. The results of the present research can be used for the further investigation of particle–fluid interactions. Full article