Generalized Bessel Polynomials Research Articles

An optimal solution for tumor growth model using generalized Bessel polynomials

In this paper, a mathematical model is given that depicts the interactions between cancer cells and viruses in the setting of oncolytic virotherapy. The model is separated into three classes, namely, concentrations of uninfected tumor cells in the population “ ”, free virus “ ”, and cancerous cells infected “ ”. Applying Caputo fractional derivative, the model is fractionalized, and using generalized Bessel polynomials, an optimal problem is solved utilizing Lagrange multipliers method. The results show that the presented method has high accuracy and is suitable for solving the nonlinear systems based on partial differential equations especially tumors models.

A new approach based on the generalized Bessel polynomials to find optimal solution of hematopoietic stem cells model

A new approach based on the generalized Bessel polynomials to find optimal solution of hematopoietic stem cells model

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

A Unified Approach to Computing the Zeros of Orthogonal Polynomials

We present a unified approach to calculating the zeros of the classical orthogonal polynomials and discuss the electrostatic interpretation and its connection to the energy minimization problem. This approach works for the generalized Bessel polynomials, including the normalized reversed variant, as well as the Viet\'e--Pell and Viet\'e--Pell--Lucas polynomials. We briefly discuss the electrostatic interpretation for each aforesaid case and some recent advances. We provide zeros and error estimates for various cases of the Jacobi, Hermite, and Laguerre polynomials and offer a brief discussion of how the method was implemented symbolically and numerically with Maple. In conclusion, we provide possible avenues for future research.

Evaluation of beam shape coefficients in T-matrix methods using a finite series technique: on blow-ups using hypergeometric functions and generalized Bessel polynomials

In T-matrix methods (generalized Lorenz–Mie theories or extended boundary condition method), beam shape coefficients encoding the shape of the illuminating structured beam have to be evaluated. This may be carried out by using the finite series technique, which, however, generates blow-ups when the partial wave order of the beam shape coefficients increases. Using hypergeometric functions and generalized Bessel polynomials, we demonstrate in the case of on-axis Gaussian beams that these blow-ups are genuine phenomena, not due to a lack of numerical precision, and we establish criteria to evaluate the critical partial wave order that implies blow-ups.

An Introductory Overview of Bessel Polynomials, the Generalized Bessel Polynomials and the q-Bessel Polynomials

Named essentially after their close relationship with the modified Bessel function Kν(z) of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θn(x) and θn(x;α,β), has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function rFs with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with rΦs which also involves r symmetric numerator parameters and s symmetric denominator parameters.

Sine-Squared Pulse Approximation for Matched Filter Design Using Generalized Bessel Polynomials and Particle Swarm Optimization

This paper presents the study of mathematical characteristics of generalized Bessel polynomial that can be applied to approximate a sine-squared pulse for designing matched filters in communication systems. The proposed pulse can be designed by using the transfer function, in which the numerator is the five pairs of a transmissions zero pairs, and the generalized Bessel polynomial is used as the denominator. A parameters of generalized Bessel polynomials can be adjusted by particle swarm optimization to find the best parameter value. From the simulation results can be found that a parameter can be adjusted. A proposed pulse is close to the ideal response in mainlobe, and one sidelobe to four sidelobes with stability, which outperformed previous research.

Optimized Gaussian Pulse Design for UWB System Using Particle Swarm Optimization Based-on Generalized Bessel Polynomials

The ultrawideband system operates a very short pulse with enormous bandwidth to provide high data rates for data transmission. To design the UWB pulse, considering the pulse shape is very necessary, and a spectral emission mask of the designed pulse should meet the FCC spectral mask requirement between frequency range 3.1 GHz to 10.6 GHz. The traditional UWB pulse design is based on the Gaussian derivative. However, the frequency spectrum is not satisfied the FCC spectral mask requirement. In this study, the Gaussian pulse can be designed from the mathematical characteristic of the generalized Bessel polynomial. The spectral efficiency of the proposed pulse can be improved by the combination of the derivative of Gaussian pulse with a weight coefficient optimization with particle swarm optimization (PSO). PSO is a population-based optimization algorithm inspired by animal behavior. PSO is applied with generalized Bessel polynomial transfer function to gain the best weight coefficient, we proposed to optimize its weight vector to design a pulse that exceeds to FCC spectral mask. The results were found in MATLAB software show that generalized Bessel polynomials can approximate the proposed pulse with combination method and PSO. The spectral efficiency is improved to 89.30% and the spectrum is greater close to the FCC spectral mask requirement. To confirm an improved spectral efficiency compared to the previous works.

Quantization of rationally deformed Morse potentials by Wronskian transforms of Romanovski-Bessel polynomials

The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including ‘juxtaposed’ pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states.

Computation of the reverse generalized Bessel polynomials and their zeros

It is well known that one of the most relevant applications of the reverse Bessel polynomials θ n ( z ) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of θ n ( z ) . In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials θ n ( z ; a ) . A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples.

New recurrence relations for several classical families of polynomials

In this paper, we derive new recurrence relations for the following families of polynomials: Nørlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol–Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.

Mathematical Model for Gaussian-Like Pulse Design for a UWB System Based on the Generalized Bessel Polynomials

Mathematical Model for Gaussian-Like Pulse Design for a UWB System Based on the Generalized Bessel Polynomials

On a generalization of a conjecture of Grosswald

We generalize a conjecture of Grosswald, now a theorem due to Filaseta and Trifonov, stating that the Bessel polynomials, denoted by yn(x), have the associated Galois group Sn over the rationals for each n. We consider generalized Bessel polynomials yn,β(x) which contain interesting families of polynomials whose discriminants are nonzero rational squares. We show that the Galois group associated with yn,β(x) always contains An if β≥0 and n sufficiently large. For β<0 the Galois group almost always contains An. It is further shown that for β<−2, under the hypothesis of the abc conjecture, the Galois group of yn,β(x) contains An for all sufficiently large n. Using these results, an earlier work of Filaseta, Finch and Leidy and the first author concerning the discriminants of yn,β(x), we are able to explicitly describe the instances where the Galois group associated with yn,β(x) is An for all sufficiently large n depending on β.

Sine-Squared pulse approximation using generalized bessel polynomials

This paper presents the approximation of sine-squared pulse based on the generalized Bessel polynomials. For designing a circuit to synthesize a sine-squared pulse test signal. The generalized Bessel polynomials have more parameter than classical Bessel polynomials that have alpha and beta parameters for adjusting the dominator of the transfer function to approximate the sine-squared pulse that closes to the ideal pulse. The simulation results show that the generalized Bessel polynomial can adjust the approximation response close to the ideal response. The orders of the transfer function are decreased that confirm a better performance than the previous works.

Some new properties of generalized Bessel polynomials

We obtain some new properties of generalized Bessel polynomials. We mainly investigate bilinear and bilateral generating function relations for these polynomials. We also derive a result giving certain families of bilateral generating functions for genera

On the construction of generalized monogenic Bessel polynomials

In recent years, much attention has been paid to the role of classical special functions of a real or complex variable in mathematical physics, especially in boundary value problems (BVPs). In the present paper, we propose a higher‐dimensional analogue of the generalized Bessel polynomials within Clifford analysis via a special set of monogenic polynomials. We give the definition and derive a number of important properties of the generalized monogenic Bessel polynomials (GMBPs), which are defined by a generating exponential function and are shown to satisfy an analogue of Rodrigues' formula. As a consequence, we establish an expansion of particular monogenic functions in terms of GMBPs and show that the underlying basic series is everywhere effective. We further prove a second‐order homogeneous differential equation for these polynomials.

Nonnegative linearization coefficients of the generalized Bessel polynomials

In this work, we solve the general linearization problem for the generalized Bessel polynomials using their inversion formula. For some particular values, we get a recurrence relation satisfied by the linearization coefficients from which we deduce their nonnegativity. We also recover a result given by Berg and Vignat (Constr Approx 27:15–32, 2008) and derived an explicit formula that generalizes a result by Atia and Zeng (Ramanujan J 28:211–221, 2012).

Riemann-hilbert characterization for main bessel polynomials with varying large negative parameters

In this article, we establish the Bessel polynomials with varying large negative parameters and discuss their orthogonality based on the generalized Bessel polynomials. By using the Riemann-Hilbert boundary value problem on the positive real axis, we get the Riemann-Hilbert characterization of the main Bessel polynomials with varying large negative parameters.

Computing the complex zeros of special functions

The complex zeros of solutions of second order linear ODEs lie over certain curves in the complex plane called anti-Stokes lines. We consider the Liouville-Green (WKB) approximation for linear homogeneous second order ODEs $$w^{\prime \prime }(z)+A(z)w(z)=0$$ with $$A(z)$$ meromorphic, and describe the qualitative properties of the approximate anti-Stokes lines. Based on this qualitative description, we construct a fourth order method which efficiently computes the complex zeros of solutions of second order ODEs following the path of the approximate anti-Stokes lines. We illustrate the method with the computation of the zeros of parabolic cylinder functions, Bessel functions and generalized Bessel polynomials and describe specific algorithms for the computation of these zeros.

A Determinant Expression for the Generalized Bessel Polynomials

Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.