 |
Discrete Mathematics & Theoretical Computer Science |
Victor J. W. Guo ; Masao Ishikawa ; Hiroyuki Tagawa ; Jiang Zeng - A generalization of Mehta-Wang determinant and Askey-Wilson polynomials
dmtcs:2337 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2337
1; Masao Ishikawa 2; Hiroyuki Tagawa 3; Jiang Zeng 4
- 1 Department of Mathematics, East China Normal University
- 2 Department of Mathematics [Okinawa]
- 3 Department of Mathematics [Wakayama]
- 4 Institut Camille Jordan
Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n×n$ determinant $\det ((a+j-i)Γ (b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\mathrm{Pf}((j-i)Γ (b+j+i))$ 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.