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Acta Physica Sinica

Binomial theorem involving Hermite polynomials and negative-binomial theorem involving Laguerre polynomials

Fan Hong-Yi, Lou Sen‐Yue, Xiao‐Yin Pan, Da Cheng

Published Jan 1, 2013• 8 citations
Publisher

Abstract

We propose an operator Hermite polynomial method, namely, we replace the arguments of the special function by quantum mechanical operators, and in this way we derive a binomial theorem involving Hermite polynomials and a negative-binomial theorem involving Laguerre polynomials. These two theorems will have essential applications in quantum optics calculations. This method is concise and helpful in deducing many operator identities, which may become a new branch in mathematical physics theory.

DOI
10.7498/aps.62.240301

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We propose an operator Hermite polynomial method, namely, we replace the arguments of the special function by quantum mechanical operators, and in this way we derive a binomial theorem involving Hermite polynomials and a negative-binomial theorem involving Laguerre polynomials. These two theorems will have essential applications in quantum optics calculations. This method is concise and helpful in deducing many operator identities, which may become a new branch in mathematical physics theory.
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By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e., By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators (IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.
Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.
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.