## A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials    [PDF]

Masao Ishikawa, Hiroyuki Tagawa, Jiang Zeng
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.
View original: http://arxiv.org/abs/1210.5305