The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polyn...The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.展开更多
In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y"-xAY’+BY=0.An explicit expression for the Hermite mat...In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y"-xAY’+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues’ formula are given.展开更多
In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix ...In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.展开更多
In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractiona...In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.展开更多
In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, thi...In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.展开更多
The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the ext...The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.展开更多
By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial(TVHP).By them and the technique of integration within an ordered prod...By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial(TVHP).By them and the technique of integration within an ordered product(IWOP) of operators we further derive new generating function formulas of the TVHP.They are useful in quantum optical theoretical calculations.It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.展开更多
In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechani...In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.展开更多
In this article,the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed.In an interesting study,Kloosterziel(Journal of Engineering Mathematics 24...In this article,the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed.In an interesting study,Kloosterziel(Journal of Engineering Mathematics 24,213-236(1990)) represented a solution to an initial value problem(IVP) for the heat equation,with an initial data in a class of rapidly decaying functions,as a series of self-similar solutions to the heat equation.This approach quickly revealed the large time behaviour for the solution to the IVP.Inspired by Kloosterziel's approach,the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation.The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.展开更多
In this article, some proper ties of complex Wiener-Ito multiple integrals and complex Ornstein-Uhlenbeck opera tors and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone form...In this article, some proper ties of complex Wiener-Ito multiple integrals and complex Ornstein-Uhlenbeck opera tors and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula, and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-Ito multiple integrals are given.展开更多
We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to ove...We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to overwhelming attractive electrodynamic potentials. The charge as a physical constant and single entity is taken as density and segments on their respective sub-quanta (floats on sub quanta) and hence the fractional charge quantiz at in. There is an integrated oscillatory effect which ties all fractional quantized states both on the surface and in the interior of the volume of an electron. The eigenfunctions, i.e., the energy profiles for the electron show the shape of a string or a quantum wire in which fractional quantized states are beaded. We followed an entirely different approach and indeed thesis to reproducing the eigenfunctions for the fractional quantized states for a single electron. We produced very fascinating mathematical formulas for all such cases by using Hermite and Laguerre polynomials, spherical based and Neumann functions and indeed asymptotic behavior of Bessel and Neumann functions. Our quantization theory is dealt in the momentum space.展开更多
General prediction formulas involving Hermite polynomials are developed for time series expressed as a transformation of a Gaussian process.The prediction gains over linear predictors are examined numerically,demonstr...General prediction formulas involving Hermite polynomials are developed for time series expressed as a transformation of a Gaussian process.The prediction gains over linear predictors are examined numerically,demonstrating the improvement of nonlinear prediction.展开更多
In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x ∈ R, t ＞ O, whereμ and k are positive constants. Since the nonhomogeneous term kx ...In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x ∈ R, t ＞ O, whereμ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) ψt - ψxx =- x2. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly.Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf' s paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.展开更多
In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares me...In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares method aid of Hermite polynomials.The suggested method reduces this type of systems to the solution of systems of linear algebraic equations.To demonstrate the accuracy and applicability of the presented method some test examples are provided.Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations.We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.展开更多
文摘The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
文摘In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y"-xAY’+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues’ formula are given.
文摘In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.
文摘In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.
文摘In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.
基金funded by Cebu Normal University through its Research Institute for Computational Mathematics and Physics(RICMP).
文摘The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.
基金supported by the National Natural Science Foundation of China (Grant No. 11174114)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 12KJD140001)the Research Foundation of Changzhou Institute of Technology of China (Grant No. YN1106)
文摘By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial(TVHP).By them and the technique of integration within an ordered product(IWOP) of operators we further derive new generating function formulas of the TVHP.They are useful in quantum optical theoretical calculations.It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.
基金Project supported by the National Natural Science Foundation of China(Grant No.11775208)the Foundation for Young Talents at the College of Anhui Province,China(Grant Nos.gxyq2021210 and gxyq2019077)the Natural Science Foundation of the Anhui Higher Education Institutions of China(Grant Nos.KJ2020A0638 and 2022AH051586)。
文摘In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.
基金supported by the Council of Scientific and Industrial Research(No.09/84(366)/2005-EMR-I)
文摘In this article,the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed.In an interesting study,Kloosterziel(Journal of Engineering Mathematics 24,213-236(1990)) represented a solution to an initial value problem(IVP) for the heat equation,with an initial data in a class of rapidly decaying functions,as a series of self-similar solutions to the heat equation.This approach quickly revealed the large time behaviour for the solution to the IVP.Inspired by Kloosterziel's approach,the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation.The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.
基金Supported by NSFC(11871079)NSFC (11731009)Center for Statistical Science,PKU
文摘In this article, some proper ties of complex Wiener-Ito multiple integrals and complex Ornstein-Uhlenbeck opera tors and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula, and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-Ito multiple integrals are given.
文摘We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to overwhelming attractive electrodynamic potentials. The charge as a physical constant and single entity is taken as density and segments on their respective sub-quanta (floats on sub quanta) and hence the fractional charge quantiz at in. There is an integrated oscillatory effect which ties all fractional quantized states both on the surface and in the interior of the volume of an electron. The eigenfunctions, i.e., the energy profiles for the electron show the shape of a string or a quantum wire in which fractional quantized states are beaded. We followed an entirely different approach and indeed thesis to reproducing the eigenfunctions for the fractional quantized states for a single electron. We produced very fascinating mathematical formulas for all such cases by using Hermite and Laguerre polynomials, spherical based and Neumann functions and indeed asymptotic behavior of Bessel and Neumann functions. Our quantization theory is dealt in the momentum space.
文摘General prediction formulas involving Hermite polynomials are developed for time series expressed as a transformation of a Gaussian process.The prediction gains over linear predictors are examined numerically,demonstrating the improvement of nonlinear prediction.
基金Beijing Natural Sciences Foundation (Grant Nos. 1992002 and 1002004) Beijing Education Committee Foundation, and partially supported by the National Youth Foundation of China.
文摘In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x ∈ R, t ＞ O, whereμ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) ψt - ψxx =- x2. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly.Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf' s paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.
文摘In this paper,we give a new numerical method for solving a linear system of fractional integro-differential equations.The fractional derivative is considered in the Caputo sense.The proposed method is least squares method aid of Hermite polynomials.The suggested method reduces this type of systems to the solution of systems of linear algebraic equations.To demonstrate the accuracy and applicability of the presented method some test examples are provided.Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations.We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.
