scholarly journals SIMULTANEOUS EIGENSTATES OF THE NUMBER-DIFFERENCE OPERATOR AND A BILINEAR INTERACTION HAMILTONIAN DERIVED BY SOLVING A COMPLEX DIFFERENTIAL EQUATION

2006 ◽  
Vol 21 (38) ◽  
pp. 2903-2911 ◽  
Author(s):  
HONG-YI FAN ◽  
WEI-BO GAO
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Difference Operator ◽  
Annihilation Operator ◽  
Hermite Polynomials ◽  
Entangled State ◽  
Complex Differential Equation ◽  
Pair Annihilation ◽  
Common Eigenvector ◽  
The Common

As a continuum work of Bhaumik et al.3 who derived the common eigenvector of the number-difference operator Q≡(a†a-b†b) and pair-annihilation operator ab, we search for the simultaneous eigenvector of Q and (ab-a†b†) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Ref. 5 in which the common eigenkets of Q and pair creators a†b† are discussed.

EINSTEIN–PODOLSKY–ROSEN ENTANGLED STATE REPRESENTATION FOR ANGULAR-MOMENTUM AND RADIUS ENTANGLEMENT

2004 ◽  
Vol 18 (07) ◽  
pp. 1043-1053 ◽  
Author(s):  
HONG-YI FAN ◽  
JUN-HUA CHEN
Keyword(s):  
Angular Momentum ◽  
Physical System ◽  
Total Momentum ◽  
Entangled State ◽  
Quantum Mechanical ◽  
Entangled State Representation ◽  
State Representation ◽  
Common Eigenvector ◽  
Einstein Podolsky Rosen ◽  
The Common

By comparison with the Einstein–Podolsky–Rosen coordinate-momentum entangled state, which is the common eigenvector of two particles' relative coordinate and total momentum, we construct a new quantum mechanical entangled state representation, namely, the entangled state representation of angular-momentum and radius. A concrete physical system which can embody the new quantum entanglement is analyzed.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

scholarly journals Hyperbolastic Models from a Stochastic Differential Equation Point of View

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1835
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Simulated Data ◽  
Real Data ◽  
Point Of View ◽  
Likelihood Method ◽  
Numerical Resolution ◽  
The Family ◽  
The Common ◽  
Linear Differential ◽  
Mean Function

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

scholarly journals Operational Methods in the Study of Sobolev-Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 124 ◽  
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Gérard Duchamp ◽  
Silvia Penson
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Umbral Calculus ◽  
Generalized Hypergeometric Functions ◽  
Normal Ordering ◽  
Gamma Functions ◽  
Image Technique ◽  
Operational Methods

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly.

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