Further results on the euler and Genocchi numbers

Abstract

Summary. We characterize the ordinary generating functions of the Genocchi and median Genoochi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.

Applications of the classical umbral calculus

“…(In combinatorial investigations, the notation H 2n+1 is usually used for what in our notation is |H 2n+1 | = (−1) n H 2n+1 .) The connection between median Genocchi numbers and the numbers F n = nE n−1 is given by the following result, due to Dumont and Zeng [12].…”

Untitled

“…Moreover, since the Poincaré polynomials of Gr f (I ) and of Gr dim P− f (P) can be easily computed, we arrive at a formula for the Poincaré polynomial (and thus for the Euler characteristic) of X . Recall (see [Feigin 2010]) that the Euler characteristic of the variety F a n+1 is given by the normalized median Genocchi number h n+1 (see [Dellac 1900;Dumont 1974;Dumont and Randrianarivony 1994;Dumont and Zeng 1994;Viennot 1982]). Using Theorem 1.2 we obtain an explicit formula for h n+1 in terms of binomial coefficients.…”

Sur les Permutations Comptées par les Nombres de Genocchi de 1-ière et 2-ième Espèce

“…du de ´ terminant de Hankel H ( k ) construit a ` partir de la suite L (0) . Or un proche parent de celui-ci a e ´ te ´ calcule ´ dans [ 3 ] , et de ce calcul re ´ sulte la proprie ´ te ´…”

“…du de ´ terminant de Hankel Z ( n ) tire ´ de la suite 1 1 2 7 38 295 и и и Ce de ´ terminant ne dif fe ` re de celui des nombres de Genocchi de 2-ie ` me espe ` ce 1 2 8 56 608 9440 и и и que par une puissance convenable de 2 . Or ce dernier a lui aussi e ´ te ´ calcule ´ , avec des notations dif fe ´ rentes , dans [ 3 ] . Par un calcul paralle ` le on e ´ tablit que…”