In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in ...In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in this paper,we prove that rooted products of some trees preserve real-rootedness of independence polynomials.In particular,we can obtain that their independence polynomials are unimodal and log-concave.展开更多
Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three d...Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.展开更多
The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with ...The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with variable length is characterized in terms of the real spectral radius of the matrix H -1 ( f 0) H (g) , where both H (f 0) and H (g) are Hurwitz like matrices. Based on this result, stability radius of control systems with interval type plants and first order controllers are determined.展开更多
Let Q n denote the class of all polynomials p(z) nonvanishing in the unit disk with deg p≤n and p (0)=1, and let W n denote the class of all polynomials s(z) satisfying deg s≤n and for all...Let Q n denote the class of all polynomials p(z) nonvanishing in the unit disk with deg p≤n and p (0)=1, and let W n denote the class of all polynomials s(z) satisfying deg s≤n and for all p∈Q n, s*p∈Q n , where * denotes the Hadamard product. Some properties for W n and Q n are obtained.展开更多
Let Q be the class of real coefficient polynomials of degree 2 with positive real part in the unit disk and constant term equal to 1. aam coefficient region of polynomials in Q is found and some sharp coefficient esti...Let Q be the class of real coefficient polynomials of degree 2 with positive real part in the unit disk and constant term equal to 1. aam coefficient region of polynomials in Q is found and some sharp coefficient estimates for the polynomials with positive real part in the unit disk are established in this paper.展开更多
Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and repr...Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.展开更多
We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f（z）f（z ...We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f（z）f（z ＋ c）.展开更多
Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r ...Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.展开更多
Let P（z） be a polynomial of degree n having no zeros in ｜z｜〈 1, then for every real or complex number β with ｜β｜≤ 1, and ｜z｜ = 1, R ≥ 1, it is proved by Dewan et al. [4] that｜P（Rz）＋β（R＋1/2）^nP（z...Let P（z） be a polynomial of degree n having no zeros in ｜z｜〈 1, then for every real or complex number β with ｜β｜≤ 1, and ｜z｜ = 1, R ≥ 1, it is proved by Dewan et al. [4] that｜P（Rz）＋β（R＋1/2）^nP（z）｜≤1/2{（｜R^n＋β（R＋1/2）^n｜＋｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜ -（｜R^n＋β（R＋1/2）^n｜-｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜}.In this paper we generalize the above inequality for polynomials having no zeros in }z} 〈 k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.展开更多
We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions. We also investigate the uniqueness problems of difference-differential p...We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions. We also investigate the uniqueness problems of difference-differential polynomials of entire functions sharing a common value.展开更多
This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists ...This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists x0 ∈ （0, 1） and a polynomial Pn∈ Fin（＋） such that ｜｜f（x）-x-x0/Pn（x）｜｜M≤Cω（f,n-1/2）M, where Пn（＋） indicates the set of all polynomials of degree n with positive coefficients展开更多
In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are of...In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.展开更多
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 prob...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.展开更多
Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G,where δ(G)=|E(G)||V(G)|2. If σ(G,x) has at least an unreal root,then G is said to be a σ-unreal gr...Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G,where δ(G)=|E(G)||V(G)|2. If σ(G,x) has at least an unreal root,then G is said to be a σ-unreal graph.Let δ(n) be the minimum edge-density over all n vertices graphs with σ-unreal roots. In this paper,by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained:For any positive integer a and rational number 0≤c≤1,there exists at least a graph sequence {G i} 1≤i≤a such that G i is σ-unreal and δ(G i)→c as n→∞ for all 1≤i≤a,and moreover, δ(n)→0 as n→∞.展开更多
In memory polynomial predistorter design, the coefficient estimation algorithm based on normalized least mean square is sensitive to initialization parameters. A predistorter based on generalized normalized gradient d...In memory polynomial predistorter design, the coefficient estimation algorithm based on normalized least mean square is sensitive to initialization parameters. A predistorter based on generalized normalized gradient descent algorithm is proposed. The merit of the GNGD algorithm is that its learning rate provides compensation for the independent assumptions in the derivation of NLMS, thus its stability is improved. Computer simulation shows that the proposed predistorter is very robust. It can overcome the sensitivity of initialization parameters and get a better linearization performance.展开更多
We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. ...We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.展开更多
In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomial...In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Delft ＆ Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior.展开更多
In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the exp...In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given.展开更多
Recently, Sun defined a new kind of refined Eulerian polynomials, namely,An(p,q)=∑π∈■np^odes(π)q^edes(π) for n ≥ 1, where Sn is the set of all permutations on {1, 2,..., n}, odes(π) and edes(π) enumerate the ...Recently, Sun defined a new kind of refined Eulerian polynomials, namely,An(p,q)=∑π∈■np^odes(π)q^edes(π) for n ≥ 1, where Sn is the set of all permutations on {1, 2,..., n}, odes(π) and edes(π) enumerate the number of descents of permutation π in odd and even positions, respectively. In this paper,we obtain an exponential generating function for An(p, q) and give an explicit formula for An(p, q)in terms of Eulerian polynomials An(q) and C(q), the generating function for Catalan numbers.In certain cases, we establish a connection between An(p, q) and An(p, 0) or An(0, q), and express the coefficients of An(0, q) by Eulerian numbers An,k. Consequently, this connection discovers a new relation between Euler numbers En and Eulerian numbers An,k.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
基金supported by the National Natural Science Foundation of China(No.12271527)。
文摘In 1987,Alavi,Malde,Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal.Although many researchers have been attracted by it,it is still open.Inspired by this conjecture,in this paper,we prove that rooted products of some trees preserve real-rootedness of independence polynomials.In particular,we can obtain that their independence polynomials are unimodal and log-concave.
基金Supported by the National Natural Science Foundation of China (Grant No.12161074)the Talent Introduction Research Foundation of Suqian University (Grant No.106-CK00042/028)+1 种基金Suqian Sci&Tech Program (Grant No.M202206)Sponsored by Qing Lan Project of Jiangsu Province and Suqian Talent Xiongying Plan of Suqian。
文摘Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.
文摘The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with variable length is characterized in terms of the real spectral radius of the matrix H -1 ( f 0) H (g) , where both H (f 0) and H (g) are Hurwitz like matrices. Based on this result, stability radius of control systems with interval type plants and first order controllers are determined.
文摘Let Q n denote the class of all polynomials p(z) nonvanishing in the unit disk with deg p≤n and p (0)=1, and let W n denote the class of all polynomials s(z) satisfying deg s≤n and for all p∈Q n, s*p∈Q n , where * denotes the Hadamard product. Some properties for W n and Q n are obtained.
文摘Let Q be the class of real coefficient polynomials of degree 2 with positive real part in the unit disk and constant term equal to 1. aam coefficient region of polynomials in Q is found and some sharp coefficient estimates for the polynomials with positive real part in the unit disk are established in this paper.
文摘Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.
基金supported by the National Natural Science Foundation of China (10871076)
文摘We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f（z）f（z ＋ c）.
文摘Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.
文摘Let P（z） be a polynomial of degree n having no zeros in ｜z｜〈 1, then for every real or complex number β with ｜β｜≤ 1, and ｜z｜ = 1, R ≥ 1, it is proved by Dewan et al. [4] that｜P（Rz）＋β（R＋1/2）^nP（z）｜≤1/2{（｜R^n＋β（R＋1/2）^n｜＋｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜ -（｜R^n＋β（R＋1/2）^n｜-｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜}.In this paper we generalize the above inequality for polynomials having no zeros in }z} 〈 k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.
基金supported by the NSFC(11026110,11101201)the NSF of Jiangxi(2010GQS0144)
文摘We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions. We also investigate the uniqueness problems of difference-differential polynomials of entire functions sharing a common value.
基金Supported by Inner Mongolia Natural Science Foundations of China (200408020108).
文摘This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists x0 ∈ （0, 1） and a polynomial Pn∈ Fin（＋） such that ｜｜f（x）-x-x0/Pn（x）｜｜M≤Cω（f,n-1/2）M, where Пn（＋） indicates the set of all polynomials of degree n with positive coefficients
文摘In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.
基金supported by NNSF of China(#11171260)RFDP of Higher Education of China(#20100141110054)
文摘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.
基金Supported by the National Natural Science Foundation of China(1 0 0 6 1 0 0 3 ) and the Science Founda-tion of the State Education Ministry of China
文摘Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G,where δ(G)=|E(G)||V(G)|2. If σ(G,x) has at least an unreal root,then G is said to be a σ-unreal graph.Let δ(n) be the minimum edge-density over all n vertices graphs with σ-unreal roots. In this paper,by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained:For any positive integer a and rational number 0≤c≤1,there exists at least a graph sequence {G i} 1≤i≤a such that G i is σ-unreal and δ(G i)→c as n→∞ for all 1≤i≤a,and moreover, δ(n)→0 as n→∞.
基金supported by the National High Technology Research and Development Program of China(2006AA01Z270).
文摘In memory polynomial predistorter design, the coefficient estimation algorithm based on normalized least mean square is sensitive to initialization parameters. A predistorter based on generalized normalized gradient descent algorithm is proposed. The merit of the GNGD algorithm is that its learning rate provides compensation for the independent assumptions in the derivation of NLMS, thus its stability is improved. Computer simulation shows that the proposed predistorter is very robust. It can overcome the sensitivity of initialization parameters and get a better linearization performance.
基金Supported by the National Natural Science Foundation of China under Grant No.11275165partly by 20140772-SIP-IPN,Mexico
文摘We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.
基金supported in part by the National Natural Science Foundation of China (10471154 and 10871212)
文摘In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Delft ＆ Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior.
文摘In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given.
基金Supported by “Liaoning Bai Qian Wan Talents Program” and by the Fundamental Research Funds for the Central Universities(Grant No.3132019323)
文摘Recently, Sun defined a new kind of refined Eulerian polynomials, namely,An(p,q)=∑π∈■np^odes(π)q^edes(π) for n ≥ 1, where Sn is the set of all permutations on {1, 2,..., n}, odes(π) and edes(π) enumerate the number of descents of permutation π in odd and even positions, respectively. In this paper,we obtain an exponential generating function for An(p, q) and give an explicit formula for An(p, q)in terms of Eulerian polynomials An(q) and C(q), the generating function for Catalan numbers.In certain cases, we establish a connection between An(p, q) and An(p, 0) or An(0, q), and express the coefficients of An(0, q) by Eulerian numbers An,k. Consequently, this connection discovers a new relation between Euler numbers En and Eulerian numbers An,k.
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.
