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Chinese Physics B 2015年第5期24卷 204-209页

作者：范洪义张鹏飞王震 Department of Physics Ningbo University Department of Material Science and Engineering University of Science and Technology of China Department of Modern Physics University of Science and Technology of China College of Science Changzhou Institute of Technology

By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are prese... 详细信息

关键词： operator-Hermite-polynomials (OHP) method generating function product of bivariate Hermite polynomials

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

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Chinese Physics B 2014年第12期23卷 30-33页

作者：展德会范洪义 Department of Material Science and Engineering University of Science and Technology of China College of Mechanic and Electronic Engineering Wuyi University

We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.

关键词： generating function two-variable Hermite polynomials Hermite polynomial method technique of integral within an ordered product of operators

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

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Chinese Physics B 2014年第6期23卷 18-22页

作者：范洪义展德会 Department of Material Science and Engineering University of Science and Technology of China

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.

关键词： generating function even- and odd-Hermite polynomials Hermite polynomial method techniqueof integral within an ordered product of operators

Joint probability generating function for degrees of active/passive random intersection graphs

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Frontiers of Mathematics in China 2012年第1期7卷 117-124页

作者： Yilun SHANG Institute for Cyber Security University of Texas at San AntonioSan AntonioTX 78249USA

Correlations of active and passive random intersection graphs are studied in this paper. We present the joint probability generating function for degrees of GactVe（n, re, p） and GPaSSiW（n, re, p）, which are genera... 详细信息

关键词： Random graph intersection graph degree generating function

On Rationality of generating function for the Number of Spanning Trees in Circulant Graphs

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Algebra Colloquium 2020年第1期27卷 87-94页

作者： A.D.Mednykh I.A.Mednykh sobolev institute of mathematics novosibirsk 630090russia Novosibirsk State University Novosibirsk 630090Russia

Let F(x)=∑∞n=1 Tsi,s2,...,sk(n)x^n be the generating function for the number,Ts1bs2,...,sk(n) of spanning trees in the circulant graph Cn(s1,S2,...,Sk).We show that F(x)is a rational function with integer coefficients satisfying the property F(x)=F(l/x).A similar result is also true for the circulant graphs C2n(s1,S2,....,Sk,n)of odd *** illustrate the obtained results by a series of examples.

关键词： spanning tree circulant graph Chebyshev polynomial generating function

INVERTING generating functionS WITH INCREASED NUMERICAL PRECISION-A COMPUTATIONAL EXPERIENCE

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Journal of Systems Science and Systems Engineering 2011年第4期20卷 475-494页

作者： Nam K. KIM Mohan L. CHAUDHRY Bong K. YOON Kilhwan KIM Department of Industrial Engineering Chonnam National University Department of Mathematics and Computer Science Royal Military College of Canada Defense Management College Korea National Defense University Internet Research Lab. Electronics and Telecommunications Research Institute

In this paper, we consider the numerical inversion of a variety of generating functions （GFs） that arise in the area of engineering and non-engineering fields. Three classes of GFs are taken into account in a comprehensive manner： classes of probability generating functions （PGFs） that are given in rational and non-rational forms, and a class of GFs that are not PGFs. Among others, those PGFs that are not explicitly given but contain a number of unknowns are largely considered as they are often encountered in many interesting applied problems. For the numerical inversion of GFs, we use the methods of the discrete （fast） Fourier transform and the Taylor series expansion. Through these methods, we show that it is remarkably easy to obtain the desired sequence to any given accuracy, so long as enough numerical precision is used in computations. Since high precision is readily available in current software packages and programming languages, one can now lift, with little effort, the so-called Laplacian curtain that veils the sequence of interest. To demonstrate, we take a series of representative examples： the PGF of the number of customers in the discrete-time Geo^X/Geo/c queue, the same in the continuous-time M^X/D/c queue, and the GFs arising in the discrete-time renewal process.

关键词： Queuing applied probability numerical inversion generating function

THE CALCULUS OF generating functionS AND THE FORMAL ENERGY FOR HAMILTONIAN ALGORITHMS

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Journal of Computational Mathematics 1998年第6期16卷 481-498页

作者： Feng K.(ICMSEC, Chinese Academy of Sciences) ICMSEC Chinese Academy of Sciences China

In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.

关键词： generating function calculus of generating functions Darboux transformation cotangent bundles Lagrangian submanifold invariance of generating function formal energy

generating functions for powers of second-order recurrence sequences

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Journal of Shanghai University(English Edition) 2011年第6期15卷 517-521页

作者：王晓霞李梅 Department of mathematics College of SciencesShanghai University

For the sequences satisfying the recurrence relation of the second order,the generating functions for the products of the powers of these sequences are *** study was from Carlita and Riordan who began a study on closed form of generating functions for powers of second-order recurrence *** investigation was completed by St*** by the recent work of Istva＇n about the non-closed generating functions of the products of the powers of the second-order sequences,the authors give several extensions of Istva＇n＇s results in this paper.

关键词： generating function second-order sequence the Binet formula

Explicit Forms of q-Deformed Lévy-Meixner Polynomials and Their generating functions

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Acta Mathematica Sinica,English Series 2008年第2期24卷 201-214页

作者： Zhi Yuan HUANG Pei Yan LI Ying WU Department of Mathematics Huazhong University of Science and Technology Wuhan 430074 P. R. China

The classical Levy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters： c1, c2,γ and β. In this paper, for-1 〈q〈 1, we obtain a unified explicit form of q-deformed Levy-Meixner polynomials and their generating functions in term of c1, c2, γand β, which is shown to be a reasonable interpolation between classical case （q=1） and fermionic case （q=-1）.In particular, when q=0 it＇s also compatible with the free case.

关键词： q-deformation Levy-Meixner polynomials generating function free probability

Some New generating functions for the Modified Laguerre Polynomials

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1398-1414页

作者： Nejla Ozmen Department of Mathematics Faculty of Science and ArtsDuzce UniversityDuzceTurkey

In this paper,we obtain some new results on bilateral generating functions of the modified Laguerre *** also get generating function relations between the modified Laguerre polynomials and the generalized Lauricella *... 详细信息

关键词： Modified Laguerre polynomials generating function multilinear and multilateral generating function recurrence relations generalized Lauricella function