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New Laguerre-polynomials’ generating functions derived by virtue of operator Hermite-polynomial method and entangled state representation

Ke Zhang, Hong-Yi Fan

Published May 1, 2025• 0 citations

Abstract

In this paper, we derive two new generating functions of Laguerre-polynomials, which look like the negative binomial theorem for the Laguerre function Lnx, by adopting the bi-partite entangled state representation and the operator-Hermite-polynomial HnX method, where X is the coordinate operator. Their application in investigating the density operator of photon loss of chaotic light is presented, and a new partition of Fock space in terms of the density operator of the negative binomial field is obtained. We also point out that the power-series definition of the Laguerre-polynomial originates from the Hermite polynomial. As a by-product, we derive a new Laguerre-polynomials-operator generating formula by virtue of the method of integration within an ordered product of operators.

DOI: 10.1063/5.0275639

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In this paper, we derive two new generating functions of Laguerre-polynomials, which look like the negative binomial theorem for the Laguerre function Lnx, by adopting the bi-partite entangled state representation and the operator-Hermite-polynomial HnX method, where X is the coordinate operator. Their application in investigating the density operator of photon loss of chaotic light is presented, and a new partition of Fock space in terms of the density operator of the negative binomial field is obtained. We also point out that the power-series definition of the Laguerre-polynomial originates from the Hermite polynomial. As a by-product, we derive a new Laguerre-polynomials-operator generating formula by virtue of the method of integration within an ordered product of operators.

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.

By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials.We also derive the new reciprocal relation of Laguerre polynomials n=0(1) n l n L n(x) = x l l!,and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated.Some new expansion identities about the operator Laguerre polynomial are also derived.This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.

In this paper, by virtue of the entangled state representation we concisely derive some new operator identities regarding to 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 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.