Application of Eigenkets of Bosonic Creation Operator in Deriving Some New Formulas of Associated Laguerre Polynomials

Using the resolution of unity composed of bosonic creation operator＇s eigenkets and annihilation operator＇s un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L^ρm （｜z｜^2） and its generalized generating function formula. A series of recursive relations regarding to L^ρm （｜z｜^2） are also deduced in the context of the Fock representation by algebraic method.

Pseudo Laguerre Matrix Polynomials, Operational Identities and Quasi-Monomiality

The main purpose of this paper is to introduce the matrix extension of the pseudo Laguerre matrix polynomials and to explore the formal properties of the operational rules and the principle of quasi-monomiality to derive a number of properties for pseudo Laguerre matrix polynomials.

Construction of Laguerre polynomial's photon-added squeezing vacuum state and its quantum properties

Laguerre polynomial’s photon-added squeezing vacuum state is constructed by operation of Laguerre polynomial’s photon-added operator on squeezing vacuum state. By making use of the technique of integration within an ordered product of operators, we derive the normalization coefficient and the calculation expression of ■. Its statistical properties, such as squeezing, the anti-bunching effect, the sub-Poissonian distribution property, the negativity of Wigner function, etc., are investigated. The influences of the squeezing parameter on quantum properties are discussed. Numerical results show that,firstly, the squeezing effect of the 1-order Laguerre polynomial’s photon-added operator exciting squeezing vacuum state is strengthened, but its anti-bunching effect and sub-Poissonian statistical property are weakened with increasing squeezing parameter; secondly, its squeezing effect is similar to that of squeezing vacuum state, but its anti-bunching effect and subPoissonian distribution property are stronger than that of squeezing vacuum state. These results show that the operation of Laguerre polynomial’s photon-added operator on squeezing vacuum state can enhance its non-classical properties.

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

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.

L^p INEQUALITIES AND ADMISSIBLE OPERATOR FOR POLYNOMIALS

Let p（z） be a polynomial of degree at most n. In this paper we obtain some new results about the dependence of on [[P（Z）[ls for every α, β∈ C with ｜a｜≤ 1, ｜β｜〉 1, R 〉 r 〉 1, and s 〉 O. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.

Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

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.

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite＇s polynomials ∫-∞＋∞（x2）mexp（-x2）{Hr（x）}2dx,∫0∞exp（-x2）H2k（x）H2s＋1（x）dx,∫0∞exp（-x2）H2k（x）H2s（x）dx and ∫0∞exp（-x2）H2k＋1（x）H2s＋1（x）dx, are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of spe- cial functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite＇s polynomials by suitable simplifications of arbitrary parameters.

Operators's-parameterized ordering and its classical correspondence in quantum optics theory

In reference to the Weyl ordering xmpn→ （1/2）m ∑l=0m （ml）Xm-lPnXl , where X and P are coordinate and momentum operator, respectively, this paper examines operators＇ s-parameterized ordering and its classical correspondence, finds the fundamental function-operator correspondence （1-s/2）（n＋m）/2Hm,n（/2/1-sα,/2/1-sα）→αman and its complementary relation anam→（-i）n＋m（1-s/2）（m＋n）/2：Hm,n（i√2/1-sa,i√2/1-sa）,where Hrn,n is the two-variable Hermite polynomial, a, at are bosonic annihilation and creation operators respectively, s is a complex parameter. The s＇-ordered operator power-series expansion of s-ordered operator atraan in terms of the two-variable Hermite polynomial is also derived. Application of operators＇ s-ordering formula in studying displaced- squeezed chaotic field is discussed.

四阶Laguerre型微分方程的左定空间和左定算子

本文讨论了某四阶Laguerre型微分方程的下有界性,并构造了相关的左定空间和左定算子.

一个偏微分方程的解与Laguerre多项式

由两种方法求解一个偏微分方程，得到了一个算子公式和关于Ｌａｇｕｅｒｒｅ多项式的两种微分表达式。

Morse势中矩阵元X_(νν′)、P_(νν′)的计算

矩阵元Ｘνν′和Ｐνν′在有关量子力学问题中有相当重要的应用，本文利用广义拉盖尔多项式的正交性和分部积分的方法，推导Ｍｏｒｓｅ 势中 Ｘνν′、Ｐνν′的计算公式．

About Classical to Quantum Weyl Correspondence

After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables. These include as well as pure powers of the amplitude as also basic periodic functions of the phase φwith their quantum-mechanical correspondence. In the representation by number states, all the considered operators involve the Jacobi polynomials as the essential formative element. Whereas the quantity in normal ordering due to its indeterminacy leads to the introduction of the notions of sub- and super-Poissonian statistics the analogous quantity in (Weyl) symmetrical orderingis positive definite and satisfies an inequality. The notions of sub- and super-Poissonian statistics are problematic when they are used for the definition of nonclassicality of states since the mentioned measure in normal ordering does not determine the Poisson statistics in their middle in unique way but determines only a large set of statistics which may be very far in the sense of the Hilbert-Schmidt distance from a Poisson statistics that is discussed.

Sumudu Transformation or What Else Can Laplace Transformation Do

The transition from a known Taylor series ?of a known function f(x) to a new function ?primarily defined by the infinite power series ?with coefficients f(n)(0)?from the Taylor series of the function f(x)?can be made by an integral transformation which is a modified Laplace transformation and is called Sumudu transformation. It makes the transition from the Exponential series to the Geometric series and may help to evaluate new infinite power series from known Taylor series. The Sumudu transformation is demonstrated to be a limiting case of Fractional integration. Apart from the basic Sumudu integral transformation we discuss a modification where the coefficients ?from the Taylor series are not changed to f(n)(0)?but only to . Beside simple examples our applications are mainly concerned to calculate new Generating functions for Hermite polynomials from the basic ones.

Wigner Quasiprobability with an Application to Coherent Phase States

Starting from Wigner’s definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables (q,p) of phase space and using the known relation to the parity operator. One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with SU (1,1) symmetry with the same index K=1/2 of unitary irreducible representations.

Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.

Generalized Eulerian Numbers

We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.

Deriving new operator identities by alternately using normally,antinormally,and Weyl ordered integration technique

Dirac's ket-bra formalism is the language of quantum mechanics.We have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors in previous work.In this work,by alternately using the technique of integration within normal,antinormal,and Weyl ordering of operators we not only derive some new operator ordering identities,but also deduce some new integration formulas regarding Laguerre and Hermite polynomials.This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique,without really performing the integrations in the ordinary way.

The s-ordered expansions of the operator function about the combined quadrature μX + νP

A general framework applicable to deriving the s-ordered operator expansions is presented in this paper.We firstly deduce the s-ordered operator expansion formula of density operator ρ a?,a and introduce the technique of integration within the sordered product of operators (IWSOP).Based on the deduction and the technique,we derive the s-ordered expansions of operators (μX + νP)n and Hn (μX + νP) (linear combinations of the coordinate operator X and the momentum operator P,Hn (x) is Hermite polynomial),respectively,and discuss some special cases of s=1,0,-1.Some new useful operator identities are obtained as well.

Cartan's Second Main Theorem and Mason's Theorem for Jackson Difference Operator

Let f:C→P^(n)be a holomorphic curve of order zero.The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves.In addition,a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial.Furthermore,they extend the Mason’s theorem for m+1 polynomials.Some examples are constructed to show that their results are accurate.