Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one ...Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）展开更多
Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds t...Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.展开更多
We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second...We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials.We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.展开更多
The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational...The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.展开更多
We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and o...The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them.展开更多
In this paper,we introduce modified degenerate polyexponential Cauchy(or poly-Cauchy)polynomials and numbers of the second kind and investigate some identities of these polynomials.We derive recurrence relations and t...In this paper,we introduce modified degenerate polyexponential Cauchy(or poly-Cauchy)polynomials and numbers of the second kind and investigate some identities of these polynomials.We derive recurrence relations and the relationship between special polynomials and numbers.Also,we introduce modified degenerate unipolyCauchy polynomials of the second kind and derive some fruitful properties of these polynomials.In addition,positive associated beautiful zeros and graphical representations are displayed with the help of Mathematica.展开更多
The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan...The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.展开更多
This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homoge...This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.展开更多
A method to estimate the probabilistic density function （PDF） of shear strength parameters was proposed. The second Chebyshev orthogonal polynomial（SCOP） combined with sample moments （the origin moments） was use...A method to estimate the probabilistic density function （PDF） of shear strength parameters was proposed. The second Chebyshev orthogonal polynomial（SCOP） combined with sample moments （the origin moments） was used to approximate the PDF of parameters. X^2 test was adopted to verify the availability of the method. It is distribution-free because no classical theoretical distributions were assumed in advance and the inference result provides a universal form of probability density curves. Six most commonly-used theoretical distributions named normal, lognormal, extreme value Ⅰ , gama, beta and Weibull distributions were used to verify SCOP method. An example from the observed data of cohesion c of a kind of silt clay was presented for illustrative purpose. The results show that the acceptance levels in SCOP are all smaller than those in the classical finite comparative method and the SCOP function is more accurate and effective in the reliability analysis of geotechnical engineering.展开更多
This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating ...This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.展开更多
The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali...The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .展开更多
Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one...Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.展开更多
基金Supported by the National Nature Science Foundation.
文摘Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）
文摘Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.
基金This work was supported by the National Research Foundation of Korea(NRF)Grant Funded by the Korea Government(No.2020R1F1A1A01071564).
文摘We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials.We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.
文摘The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.
文摘We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
基金supported by the Basic Science Research Program,the National Research Foundation of Korea(NRF-2021R1F1A1050151).
文摘The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them.
基金supported by the Taif University Researchers Supporting Project(TURSP-2020/246),Taif University,Taif,Saudi Arabia.
文摘In this paper,we introduce modified degenerate polyexponential Cauchy(or poly-Cauchy)polynomials and numbers of the second kind and investigate some identities of these polynomials.We derive recurrence relations and the relationship between special polynomials and numbers.Also,we introduce modified degenerate unipolyCauchy polynomials of the second kind and derive some fruitful properties of these polynomials.In addition,positive associated beautiful zeros and graphical representations are displayed with the help of Mathematica.
文摘The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.
基金Hunan Provincial Natural Science Foundation Under Grant No.02JJY2085
文摘This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.
基金Projects(50490274 , 10472134 , 50404010) supported by the National Natural Science Foundation of China project(2002CB412703) supported by the Key Fundamental Research and Development Programof China
文摘A method to estimate the probabilistic density function （PDF） of shear strength parameters was proposed. The second Chebyshev orthogonal polynomial（SCOP） combined with sample moments （the origin moments） was used to approximate the PDF of parameters. X^2 test was adopted to verify the availability of the method. It is distribution-free because no classical theoretical distributions were assumed in advance and the inference result provides a universal form of probability density curves. Six most commonly-used theoretical distributions named normal, lognormal, extreme value Ⅰ , gama, beta and Weibull distributions were used to verify SCOP method. An example from the observed data of cohesion c of a kind of silt clay was presented for illustrative purpose. The results show that the acceptance levels in SCOP are all smaller than those in the classical finite comparative method and the SCOP function is more accurate and effective in the reliability analysis of geotechnical engineering.
文摘This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.
文摘The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .
基金Supported by the National Natural Science Foundation of China (Grant No. 10601065)
文摘Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.
