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Register a new userA generalization of the Mehta-Wang determinant and Askey-Wilson polynomials
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Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $det((a+j-i)Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $Pf((j-i)Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.
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We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newtons interpolation formula.
In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic interpolation formulas by the methods of matrix
We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.
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