The median Genocchi numbers, qq-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 qq-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 qq-version of the normalized median Genocchi numbers introduced by Han and Zeng.

► We describe the Poincaré polynomials of the degenerate flag varieties via weighted Motzkin paths.
► We derive the continued fraction form for the generating function of the qq-Genocchi numbers.
► We identify the Poincaré polynomials with the Han–Zeng qq-Genocchi numbers.