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Register a new userBi-orthogonal Polynomials and the Five parameter Asymmetric Simple Exclusion Process
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We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $mathbf{d}$ and $mathbf{e}$ -- for the five parameter ($alpha$, $beta$, $gamma$, $delta$ and $q$) Asymmetric Simple Exclusion Process. This method requires an $LDU$ decomposition of the ``bi-moment matrix. The decomposition defines a new pair of basis vectors sets, the `boundary basis. This basis is defined by the action of polynomials ${P_n}$ and ${Q_n}$ on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie. each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed `bi-orthogonal. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator $mathbf{d}+mathbf{e}$ is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).
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Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. Recurrence relations are shown for the Lie groups of types $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.
We investigate a class of power series occurring in some problems in quantum optics. Their coefficients are either Gegenbauer or Laguerre polynomials multiplied by binomial coefficients. Although their sums have been known for a long time, we employ here a different method to recover them as higher-order derivatives of the generating function of the given orthogonal polynomials. The key point in our proof consists in exploiting a specific functional equation satisfied by the generating function in conjunction with Cauchys integral formula for the derivatives of a holomorphic function. Special or limiting cases of Gegenbauer polynomials include the Legendre and Chebyshev polynomials. The series of Hermite polynomials is treated in a straightforward way, as well as an asymptotic case of either the Gegenbauer or the Laguerre series. Further, we have succeeded in evaluating the sum of a similar power series which is a higher-order derivative of Mehlers generating function. As a prerequisite, we have used a convenient factorization of the latter that enabled us to employ a particular Laguerre expansion. Mehlers summation formula is then applied in quantum mechanics in order to retrieve the propagator of a linear harmonic oscillator.
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