On an extension of the Genocchi numbers (Q1891364)

The authors investigate the sequence of polynomials \(B_ n(x, y)\) defined by the recurrence \[ B_ 1(x, y)= 1,\;B_ n(x, y)= (x+ 1)(y+ 1) B_{n- 1}(x+ 1, y+ 1)- xyB_{n- 1}(x, y) \] which was introduced initially by Stieltjes and mentioned by Touchard. This sequence turns out to be a generalization of the Gandhi polynomials [\textit{J. M. Gandhi}, Am. Math. Mon. 77, 505-506 (1970; Zbl 0198.370)] which generate the Genocchi numbers. A generating function for \(B_ n(x, y)\) is obtained, including a representation by a continued fraction. Another expression of \(B_ n(x, y)\) is given in terms of maximal points of surjective mappings of the sequence \((1, 2,\dots, 2n)\). For related problems see the same authors [Discrete Math. 132, No. 1-3, 37-49 (1994; Zbl 0807.05001)].