In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. ...In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.展开更多
Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w...Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w); |w|>1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, L n(·) means the leading coefficient of the orthonormal matrix polynomials Φ n(z;·). Finally, we deduce the asymptotic behavior of Φ n(w;)Φ n(w;Ω)* in the case when M=I.展开更多
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact ...In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.展开更多
The primary purpose of this paper is to present the Volterra integral equa- tion of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.
In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present se...In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present several recurrence relations of degenerateλ-array type polynomials and numbers.展开更多
The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other...The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.展开更多
An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial ...An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.展开更多
A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a map...A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.展开更多
In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for...In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.展开更多
From the basic properties of skein systems, we build a generalized tangle algebra (GTA). The elements of GTA are four basic tangles. There are three operations, which are connection, splicing and scalar multiplication...From the basic properties of skein systems, we build a generalized tangle algebra (GTA). The elements of GTA are four basic tangles. There are three operations, which are connection, splicing and scalar multiplication. From GTA we derive two generalized recursion formulae (GRF) and prove the existence of a generalized skein relation which satisfies GRF. The obtained generalized skein relation epitomizes all generalizations from the Jones polynomial and thus forms a unified model. Two important topological parameters, twisting measure and loop values, appear explicitly in the expressions of the unified model, and this fact greatly simplifies the operations.展开更多
The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only...The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.展开更多
Linear dispersion relation for linear wave and a Kadomtsev-Petviashvili (KP) equation for nonlinearwave are given for the unmagnetized two-ion-temperature cold dusty plasma with many different dust grain species.The n...Linear dispersion relation for linear wave and a Kadomtsev-Petviashvili (KP) equation for nonlinearwave are given for the unmagnetized two-ion-temperature cold dusty plasma with many different dust grain species.The numerical results of variations of linear dispersion with respect to the different dust size distribution are given.Moreover,how the amplitude,width,and propagation velocity of solitary wave vary vs different dust size distribution isalso studied numerically in this paper.展开更多
A global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach is presented,with respect to the polynomial degree.The domains of uniformity are described in certain phase variables.A resurgenc...A global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach is presented,with respect to the polynomial degree.The domains of uniformity are described in certain phase variables.A resurgence relation within the sequence of Riemann-Hilbert problems is observed in the procedure of derivation.Global asymptotic approximations are obtained in terms of the Airy function.The system of Hermite polynomials is used as an illustration.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
We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produc...We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produced, if the term orders ＇ and ＇ ＇ are difference-differential degree compatibility. So we present a condition on the generalized orders and prove that under the condition the algorithm for computing relative GrSbner bases will terminate. Also the relative Gr6bner bases exist under the condition. Finally, we prove the algorithm for computation of the bivariate dimension polynomials in difference-differential modules terminates.展开更多
文摘In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.
文摘Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w); |w|>1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, L n(·) means the leading coefficient of the orthonormal matrix polynomials Φ n(z;·). Finally, we deduce the asymptotic behavior of Φ n(w;)Φ n(w;Ω)* in the case when M=I.
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
文摘In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.
文摘The primary purpose of this paper is to present the Volterra integral equa- tion of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.
基金The first two authors,Mrs.Lan Wu and Xue-Yan Chen,were partially supported by the College Scientific Research Project of Inner Mongolia(Grant No.NJZY19156 and Grant No.NJZZ19144)by the Natural Science Foundation Project of Inner Mongolia(Grant No.2021LHMS05030)by the Development Plan for Young Technological Talents in Colleges and Universities of Inner Mongolia(Grant No.NJYT22051)in China.
文摘In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present several recurrence relations of degenerateλ-array type polynomials and numbers.
文摘The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.
基金the National Natural Science Foundation of China (Nos. 50679097 and 50778184).
文摘An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.
基金the Major Basic Project of China(Grant No.2005CB321702)the National Natural Science Foundation of China(Grant Nos.10431050,60573023)
文摘A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.
基金supported by the National Natural Science Foundation of China under Grant Nos.11971341 and 12001492the Natural Science Foundation of Zhejiang Province under Grant No.LQ20A010004.
文摘In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.
文摘From the basic properties of skein systems, we build a generalized tangle algebra (GTA). The elements of GTA are four basic tangles. There are three operations, which are connection, splicing and scalar multiplication. From GTA we derive two generalized recursion formulae (GRF) and prove the existence of a generalized skein relation which satisfies GRF. The obtained generalized skein relation epitomizes all generalizations from the Jones polynomial and thus forms a unified model. Two important topological parameters, twisting measure and loop values, appear explicitly in the expressions of the unified model, and this fact greatly simplifies the operations.
基金Supported by the NSFC (10771058, 11071062, 10871205), NSFH (10JJ3065)Scientific Research Fund of Hunan Provincial Education Department (10A033)Hunan Provincial Degree and Education of Graduate Student Foundation (JG2009A017)
文摘The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.
基金Supported by the National Natural Science Foundation of China under Grant No.10875082the Natural Science Foundation of Gansu Province under Grant No.3ZS061-A25-013the Natural Science Foundation of Northwest Normal University under Grant No.NWNUKJCXGC-03-17,03-48
文摘Linear dispersion relation for linear wave and a Kadomtsev-Petviashvili (KP) equation for nonlinearwave are given for the unmagnetized two-ion-temperature cold dusty plasma with many different dust grain species.The numerical results of variations of linear dispersion with respect to the different dust size distribution are given.Moreover,how the amplitude,width,and propagation velocity of solitary wave vary vs different dust size distribution isalso studied numerically in this paper.
基金supported in part by National Natural Science Foundation of China(Grant Nos. 10471154,10871212)
文摘A global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach is presented,with respect to the polynomial degree.The domains of uniformity are described in certain phase variables.A resurgence relation within the sequence of Riemann-Hilbert problems is observed in the procedure of derivation.Global asymptotic approximations are obtained in terms of the Airy function.The system of Hermite polynomials is used as an illustration.
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Acknowledgements The authors thank the anonymous referees for their constructive comments and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11271040）.
文摘We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produced, if the term orders ＇ and ＇ ＇ are difference-differential degree compatibility. So we present a condition on the generalized orders and prove that under the condition the algorithm for computing relative GrSbner bases will terminate. Also the relative Gr6bner bases exist under the condition. Finally, we prove the algorithm for computation of the bivariate dimension polynomials in difference-differential modules terminates.
