In this paper, we mainly study the uniqueness of specific q-shift difference polynomials and of meromorphic functions, which share a common small function and get the corresponding results. In addition, we also invest...In this paper, we mainly study the uniqueness of specific q-shift difference polynomials and of meromorphic functions, which share a common small function and get the corresponding results. In addition, we also investigate the problem of value distribution on q-shift difference polynomials of entire functions.展开更多
The first purpose of this paper is to study the properties on some q-shift difference differential polynomials of meromorphic functions,some theorems about the zeros of some q-shift difference-differential polynomials...The first purpose of this paper is to study the properties on some q-shift difference differential polynomials of meromorphic functions,some theorems about the zeros of some q-shift difference-differential polynomials with more general forms are obtained.The second purpose of this paper is to investigate the properties on the Nevanlinna deficiencies for q-shift difference differential monomials of meromorphic functions,we obtain some relations amongδ(∞,f),δ(∞,f′),δ(∞,f(z)^(n)f(qz+c)^(m)f′(z)),δ(∞,f(qz+c);f′(z))andδ(∞,f(z)^(n)f(qz+c)^(m)).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
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
In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some exam...In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.展开更多
Let f(z) be a function transcendental and meromorphic in the plane of growth order less than 1. This paper focuses on discuss and estimate the number of the zeros of a certain homogeneous difference polynomials of deg...Let f(z) be a function transcendental and meromorphic in the plane of growth order less than 1. This paper focuses on discuss and estimate the number of the zeros of a certain homogeneous difference polynomials of degree k in f(z), and obtains that this certain homogeneous difference polynomials has infinitely many zeros.展开更多
In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely ma...In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-af^(n)(z) and f (z)f (z+c_1)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.展开更多
In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithm...In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.展开更多
Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the...Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.展开更多
The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑...The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑n=0^F fn^tn. This equation is investigated with the use of the method of polynomial quasisolutions based on the representation of an unknown function in the form of polynomial x（t） = ∑n=0^N xn^tn. As a result of substitution of this function into equation （＊）, there appears a residual △（t） = 0（t^N）, for which an exact analytical representation has been obtained. In turn, this allows one to find the unknown coefficients xn and consequently the polynomial quasisolution x（t）. Several examples are considered.展开更多
In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f...In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f of degree d(≤n-2),p1,p2 are small meromorphic functions of e^(z),andα1,α2 are nonzero constants.Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order.As its applications,we also find some type of nonlinear difference equations having no finite order entire solutions.展开更多
A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and c...A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.展开更多
In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. Th...In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.展开更多
In this paper I present a novel polynomial regression method called Finite Difference Regression for a uniformly sampled sequence of noisy data points that determines the order of the best fitting polynomial and provi...In this paper I present a novel polynomial regression method called Finite Difference Regression for a uniformly sampled sequence of noisy data points that determines the order of the best fitting polynomial and provides estimates of its coefficients. Unlike classical least-squares polynomial regression methods in the case where the order of the best fitting polynomial is unknown and must be determined from the R2 value of the fit, I show how the t-test from statistics can be combined with the method of finite differences to yield a more sensitive and objective measure of the order of the best fitting polynomial. Furthermore, it is shown how these finite differences used in the determination of the order, can be reemployed to produce excellent estimates of the coefficients of the best fitting polynomial. I show that not only are these coefficients unbiased and consistent, but also that the asymptotic properties of the fit get better with increasing degrees of the fitting polynomial.展开更多
In this paper,we introduce the △_(h)-Gould-Hopper Appell polynomials An(x,y;h)via h-Gould-Hopper polynomials G_(n)^(h)(x,y).These polynomials reduces to △_(h)-Appell polynomials in the case y=0,△-Appell polynomials...In this paper,we introduce the △_(h)-Gould-Hopper Appell polynomials An(x,y;h)via h-Gould-Hopper polynomials G_(n)^(h)(x,y).These polynomials reduces to △_(h)-Appell polynomials in the case y=0,△-Appell polynomials in the case y=0 and h=2D-Appell polynomials in the case h→0,2D △-Appell polynomials in the case h=1 and Appell polynomials in the case h→0 and y=0.We obtain some well known main properties and an explicit form,determinant representation,recurrence relation,shift operators,difference equation,integro-difFerence equation and partial difference equation satisfied by them.Determinants satisfied by △_(h)-Gould-Hopper Appell polynomials reduce to determinant of all subclass of the usual polynomials.Recurrence,shift operators and difference equation satisfied by these polynomials reduce to recurrence,shift operators and difference equation of △_(h)-Appell polynomials,△-Appell polynomials;recurrence,shift operators,differential and integro-differential equation of 2D-Appell polynomials,recurrence,shift operators,integrodifference equation of 2D △-Appell polynomials,recurrence,shift operators,differential equation of Appell polynomials in the corresponding cases.In the special cases of the determining functions,we present the explicit forms,determinants,recurrences,difference equations satisfied by the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Carlitz Euler polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind.In particular cases of the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind,corresponding determinants,recurrences,shift operators and difference equations reduce to all subclass of degenerate so-called families except for Genocchi polynomials recurrence,shift operators,and differential equation.Degenerate Gould-Hopper Carlitz Euler polynomials do not satisfy the recurrences and differential equations of 2D-Euler and Euler polynomials.展开更多
A fundamental algebraic relationship for a general polynomial of degree n is given and proven by mathematical induction. The stated relationship is based on the well-known property of polynomials that the nth-differen...A fundamental algebraic relationship for a general polynomial of degree n is given and proven by mathematical induction. The stated relationship is based on the well-known property of polynomials that the nth-differences of the subsequent values of an nth-order polynomial are constant.展开更多
In 1998, Maschietti constructed several cyclic difference sets from monomial hyperovals. R. Evans, H.D.L. Holloman, C. Krattnthaler and Qing Xiang gave an algebraic proof of the two autocorrelation property of the rel...In 1998, Maschietti constructed several cyclic difference sets from monomial hyperovals. R. Evans, H.D.L. Holloman, C. Krattnthaler and Qing Xiang gave an algebraic proof of the two autocorrelation property of the related binary sequence. In this paper, we show that hyperovals are very closely related to two-to-one maps, and then we proceed to generalize Maschietti＇s result.展开更多
Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is k...Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.展开更多
In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. T...In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro- differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O（hm＋l） as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.展开更多
With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and gen...With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and generalize some recent results due to Meng(Math.Bohem.139:89-97,2014).展开更多
文摘In this paper, we mainly study the uniqueness of specific q-shift difference polynomials and of meromorphic functions, which share a common small function and get the corresponding results. In addition, we also investigate the problem of value distribution on q-shift difference polynomials of entire functions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11761035,12161074)the Natural Science Foundation of Jiangxi Province(Grant No.20181BAB201001)the Foundation of Education Department of Jiangxi Province(Grant Nos.GJJ190876,GJJ202303,GJJ201813,GJJ191042)。
文摘The first purpose of this paper is to study the properties on some q-shift difference differential polynomials of meromorphic functions,some theorems about the zeros of some q-shift difference-differential polynomials with more general forms are obtained.The second purpose of this paper is to investigate the properties on the Nevanlinna deficiencies for q-shift difference differential monomials of meromorphic functions,we obtain some relations amongδ(∞,f),δ(∞,f′),δ(∞,f(z)^(n)f(qz+c)^(m)f′(z)),δ(∞,f(qz+c);f′(z))andδ(∞,f(z)^(n)f(qz+c)^(m)).
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）.
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 the National Natural Science Foundation of China(10771121,11301220,11371225)the Tianyuan Fund for Mathematics(11226094)+2 种基金the NSF of Shandong Province,China(ZR2012AQ020,ZR2010AM030)the Fund of Doctoral Program Research of Shaoxing College of Art and Science(20135018)the Fund of Doctoral Program Researchof University of Jinan(XBS1211)
文摘In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.
文摘Let f(z) be a function transcendental and meromorphic in the plane of growth order less than 1. This paper focuses on discuss and estimate the number of the zeros of a certain homogeneous difference polynomials of degree k in f(z), and obtains that this certain homogeneous difference polynomials has infinitely many zeros.
基金Supported by the National Natural Science Foundation of China (11926201)Natural Science Foundation of Guangdong Province (2018A030313508)。
文摘In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-af^(n)(z) and f (z)f (z+c_1)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.
文摘In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.
文摘Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.
文摘The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑n=0^F fn^tn. This equation is investigated with the use of the method of polynomial quasisolutions based on the representation of an unknown function in the form of polynomial x（t） = ∑n=0^N xn^tn. As a result of substitution of this function into equation （＊）, there appears a residual △（t） = 0（t^N）, for which an exact analytical representation has been obtained. In turn, this allows one to find the unknown coefficients xn and consequently the polynomial quasisolution x（t）. Several examples are considered.
基金supported by the National Natural Science Foundation of China(11661044)
文摘In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f of degree d(≤n-2),p1,p2 are small meromorphic functions of e^(z),andα1,α2 are nonzero constants.Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order.As its applications,we also find some type of nonlinear difference equations having no finite order entire solutions.
文摘A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.
文摘In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.
文摘In this paper I present a novel polynomial regression method called Finite Difference Regression for a uniformly sampled sequence of noisy data points that determines the order of the best fitting polynomial and provides estimates of its coefficients. Unlike classical least-squares polynomial regression methods in the case where the order of the best fitting polynomial is unknown and must be determined from the R2 value of the fit, I show how the t-test from statistics can be combined with the method of finite differences to yield a more sensitive and objective measure of the order of the best fitting polynomial. Furthermore, it is shown how these finite differences used in the determination of the order, can be reemployed to produce excellent estimates of the coefficients of the best fitting polynomial. I show that not only are these coefficients unbiased and consistent, but also that the asymptotic properties of the fit get better with increasing degrees of the fitting polynomial.
文摘In this paper,we introduce the △_(h)-Gould-Hopper Appell polynomials An(x,y;h)via h-Gould-Hopper polynomials G_(n)^(h)(x,y).These polynomials reduces to △_(h)-Appell polynomials in the case y=0,△-Appell polynomials in the case y=0 and h=2D-Appell polynomials in the case h→0,2D △-Appell polynomials in the case h=1 and Appell polynomials in the case h→0 and y=0.We obtain some well known main properties and an explicit form,determinant representation,recurrence relation,shift operators,difference equation,integro-difFerence equation and partial difference equation satisfied by them.Determinants satisfied by △_(h)-Gould-Hopper Appell polynomials reduce to determinant of all subclass of the usual polynomials.Recurrence,shift operators and difference equation satisfied by these polynomials reduce to recurrence,shift operators and difference equation of △_(h)-Appell polynomials,△-Appell polynomials;recurrence,shift operators,differential and integro-differential equation of 2D-Appell polynomials,recurrence,shift operators,integrodifference equation of 2D △-Appell polynomials,recurrence,shift operators,differential equation of Appell polynomials in the corresponding cases.In the special cases of the determining functions,we present the explicit forms,determinants,recurrences,difference equations satisfied by the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Carlitz Euler polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind.In particular cases of the degenerate Gould-Hopper Carlitz Bernoulli polynomials,degenerate Gould-Hopper Genocchi polynomials,△_(h)-Gould-Hopper Boole polynomials and △_(h)-Gould-Hopper Bernoulli polynomials of the second kind,corresponding determinants,recurrences,shift operators and difference equations reduce to all subclass of degenerate so-called families except for Genocchi polynomials recurrence,shift operators,and differential equation.Degenerate Gould-Hopper Carlitz Euler polynomials do not satisfy the recurrences and differential equations of 2D-Euler and Euler polynomials.
文摘A fundamental algebraic relationship for a general polynomial of degree n is given and proven by mathematical induction. The stated relationship is based on the well-known property of polynomials that the nth-differences of the subsequent values of an nth-order polynomial are constant.
文摘In 1998, Maschietti constructed several cyclic difference sets from monomial hyperovals. R. Evans, H.D.L. Holloman, C. Krattnthaler and Qing Xiang gave an algebraic proof of the two autocorrelation property of the related binary sequence. In this paper, we show that hyperovals are very closely related to two-to-one maps, and then we proceed to generalize Maschietti＇s result.
基金supported by the China Scholarship Council (CSC)supported in part by the Academy of Finland #121281
文摘Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.
文摘In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro- differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O（hm＋l） as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.
文摘With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and generalize some recent results due to Meng(Math.Bohem.139:89-97,2014).
