This paper applies the convolutional differentiator method, based on generalized Forsyte orthogonal polynomial （CFPD）, to simulate the seismic wave propagation in two-phase media. From the numerical results we can s...This paper applies the convolutional differentiator method, based on generalized Forsyte orthogonal polynomial （CFPD）, to simulate the seismic wave propagation in two-phase media. From the numerical results we can see that three types of waves, fast P-waves, S-waves and slow P-waves, can be observed in the seismic wave field. The experiments on anisotropic models demonstrate that the wavefront is elliptic instead of circular and S-wave splitting occurs in anisotropic two-phase media. The research has confirmed that the rules of elastic wave propagation in fluid-saturated porous media are controlled by Biot＇s theory. Experiment on a layered fault model shows the wavefield generated by the interface and the fault very well, indicating the effectiveness of CFPD method on the wavefield modeling for real layered media in the Earth. This research has potential applications to the investigation of Earth＇s deep structure and oil/gas exploration.展开更多
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
Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.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
基金supported by the National Natural Science Foundation of China(Grant No.40874045)Special Funds for Sciences and Technology Research of Public Welfare Trades(Grant Nos. 200811021 and 201011042)
文摘This paper applies the convolutional differentiator method, based on generalized Forsyte orthogonal polynomial （CFPD）, to simulate the seismic wave propagation in two-phase media. From the numerical results we can see that three types of waves, fast P-waves, S-waves and slow P-waves, can be observed in the seismic wave field. The experiments on anisotropic models demonstrate that the wavefront is elliptic instead of circular and S-wave splitting occurs in anisotropic two-phase media. The research has confirmed that the rules of elastic wave propagation in fluid-saturated porous media are controlled by Biot＇s theory. Experiment on a layered fault model shows the wavefield generated by the interface and the fault very well, indicating the effectiveness of CFPD method on the wavefield modeling for real layered media in the Earth. This research has potential applications to the investigation of Earth＇s deep structure and oil/gas exploration.
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Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
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.
