Basic Hypergeometric Functions and Orthogonal Laurent Polynomials
نویسندگان
چکیده
A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2Φ1(q−n, qb+1; q−c+b−n; q, qz)}n=0, where 0 < q < 1 and the complex parameters b, c and d are such that b = −1,−2, . . ., c− b+ 1 = −1,−2, . . ., Re(d) > 0 and Re(c− d+ 2) > 0. Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.
منابع مشابه
Orthogonal basic hypergeometric Laurent polynomials
The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric AskeyWilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3’s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3’s which are Laurent polynomials in...
متن کاملElliptic hypergeometric Laurent biorthogonal polynomials with a dense point spectrum on the unit circle
We construct new elliptic solutions of the qd-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function 3G2(z). Their recurrence coefficients are expressed in terms of the elliptic functions. 1991 Mathematics Subject ...
متن کاملA q-SAMPLING THEOREM AND PRODUCT FORMULA FOR CONTINUOUS q-JACOBI FUNCTIONS
In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.
متن کاملSzegő Polynomials from Hypergeometric Functions
Szegő polynomials with respect to the weight function ω(θ) = eηθ [sin(θ/2)]2λ, where η, λ ∈ R and λ > −1/2 are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.
متن کاملThe Askey Scheme for Hypergeometric Orthogonal Polynomials Viewed from Asymptotic Analysis
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.
متن کامل