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

Victor J. W. Guo 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 ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : 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.
Keywords : The Mehta-Wang determinants the moments of the little q-Jacobi polynomials the Askey-Wilson polynomials.
Type de document :
Communication dans un congrès
Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.719-730, 2013, DMTCS Proceedings
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Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa, Jiang Zeng. A generalization of Mehta-Wang determinant and Askey-Wilson polynomials. Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.719-730, 2013, DMTCS Proceedings. 〈hal-01229667〉

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