The associated Askey-Wilson polynomials

Abstract

Abstract. We derive some contiguous relations for very well-poised 8<^7 series and use them to construct two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials. We then use these solutions to find explicit representations of two families of associated Askey-Wilson polynomials. We identify the corresponding continued fractions as quotients of two very well-poised 8^>7 series and find the weight functions.

Multivariable Askey–Wilson function and bispectrality

“…Thus, we can use (2.4) to define p n (z) for arbitrary n ∈ C such that |adq n−1 | > 1 and we can use analytic continuation to prove that (2.2) still holds. As we explained in the introduction, this fact is well known and goes back to the works [12,14,19].…”

“…The fact that the multivariable Askey-Wilson function satisfies difference equations amounts to certain contiguous relation between products of very-well-poised 8 φ 7 series. In the one-dimensional case similar identities were used in [12]. When q → 1, the 8 φ 7 's reduce to 7 F 6 's and one obtains meromorphic extensions of the multivariable Wilson polynomials considered by Tratnik [21].…”

Determinant solutions of the nonautonomous discrete Toda equation associated with the deautonomized discrete KP hierarchy

“…whose new entries m 0,j and m j,0 satisfy (22) with φ 0,n = φ n and φ * 0,n = φ * n for j = 1, 2, · · · , N. Proposition 5 The Gram-type determinants defined by (21), (24), and (25) satisfy the bilinear equations (16) and (20).…”

Linearization formula for the product of associated q-ultraspherical polynomials

“…Based on Theorem 1, Ismail and Rahman [6] gave a brief discussion of the nonnegativity of the linearization coefficients for the associated Askey-Wilson polynomials. Linearization of the product of the symmetric orthogonal polynomials, of which ultraspherical, associated ultraspherical and their q-analogues are examples, was discussed by Markett [7].…”