Evaluation of a certain q-determinant

A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials

“…which can be considered as the recurrence relation for a special case of the Meixner-Pollaczek polynomials (see [14,15]), and one may notice that the sequence {Γ(b + n)} n≥0 of the Gamma functions in the left-hand side can be considered as the moment sequence of the Laguerre polynomials (see, for example, [10,11,17]). Nishizawa [15] obtains a q-analogue of (1.1), which will be stated below.…”

“…which can be considered as the recurrence relation for a special case of the Meixner-Pollaczek polynomials (see [14,15]), and one may notice that the sequence {Γ(b + n)} n≥0 of the Gamma functions in the left-hand side can be considered as the moment sequence of the Laguerre polynomials (see, for example, [10,11,17]). Nishizawa [15] obtains a q-analogue of (1.1), which will be stated below. Here we replace the Gamma functions by the moments of the little q-Jacobi polynomials and show that we obtain a special case of the Askey-Wilson polynomials as D n , which also generalize the two results in our previous papers [6,7].…”

“…This identity enable us to prove Theorem 2.1 by induction. First we cite the q-binomial formula [4,10,11,16] (see also [15,Lemma 2]). Proposition 2.2.…”

A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials

“…The following formula is an extension of Nishizawa's q-analogue of Mehta-Wang's formula [16,18,8]. ; q, q .…”

“…Our main result is the following quadratic formula for basic hypergeometric series, which was discovered by applying Desnanot-Jacobi adjoint matrix theorem to compute some determinants of Mehta-Wang type [16,18,14,8] or deformed Gram determinants of orthogonal polynomials [20].…”