Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))

Abstract

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra U_q(su(1,1)) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial U_q(su(1,1))-basis and the basis of eigenfunctions are interrelated by a matrix with entries, expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials M_n(x;b,c;q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

15 pages, LaTeX

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Realizations of $su(1,1)$ and $U_q(su(1,1))$ and generating functions for orthogonal polynomials
• On the spectrum of a Hamiltonian defined on su_q(2) and quantum optical models

Cited by in corpus (14)

• Orthogonal Polynomials from Hermitian Matrices
• The Universal Askey-Wilson Algebra
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Orthogonal Polynomials from Hermitian Matrices II
• Duality of q-polynomials, orthogonal on countable sets of points
• q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
• Exactly solvable discrete time Birth and Death processes
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras
• Markov Chains Generated by Convolutions of Orthogonality Measures
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• A set of orthogonal polynomials, dual to alternative q-Charlier polynomials
• An algebraic interpretation of the $q$-Meixner polynomials
• Hamiltonian Type Operators in Representations of the Quantum Algebra U_q(su_{1,1})
• Certain generating functions for Cigler's polynomials