Result: Further results on the euler and Genocchi numbers: Further results on the Euler and Genocchi numbers

The Genocchi number \(G_{2n}\) is defined by \(t + \sum_{n \geq 1} (- 1)^ nG_{2n} {t^{2n} \over(2n)!} = {2t \over e^ t+1}\). The median Genocchi number \(H_{2n+1}\) is defined by \(H_{2n+1} = (-1)^ ng_ n^{n+1}\) where \(g_ n^ k = g_ n^{k-1} + g_{n+1}^{k-1}\), \(g^ 0_{2n} = (-1)^ nG_{2n}\), \(g^ 0_{2n+1} = 0\). The generating functions \(g(x)\) and \(h(x)\) of these numbers are characterized as the unique solutions of the functional equations \(g(x) + g({x \over 1- x}) = 2x^ 2\) and \(h({x^ 2 \over 1 + x}) + h({x^ 2 \over 1-x}) = 2x^ 2\), and connections with the Euler number such as \(2^{2n} H_{2n+1} = \sum^ n_{m = 0} (2m+1) {n \choose m} E_{2m}\) are obtained. Direct algebraic proofs of several continued fractions for \(g(x)\) and \(h(x)\) are also obtained.