The median Genocchi numbers, -analogues and continued fractions

It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The q -analogues of the Genocchi numbers can be naturally defined as the Poincaré polynomials of the degenerate flag varieties. We prove that the generating function of the Poincaré polynomials can be written as a simple continued fraction. As an application we prove that the Poincaré polynomials coincide with the q -version of the normalized median Genocchi numbers introduced by Han and Zeng.