LINEAR RECURSIVE RELATIONS FOR BERNOULLI NUMBERS AND APPLICATIONS

This study concerns a new approach of Bernoulli numbers $B_n$ and Bernoulli numbers $B_n^{(k)}$ of order $k\geq 2$, using properties of some linear recursive relations of infinite order. Linear recursive relations for generating $B_n$ and $B_n^{(k)}$ are established and some identities are provided. Moreover, linear, combinatorial and analytic approaching processes of $B_n$ and $B_n^{(k)}$ are proposed. The closed connection with partial Bell polynomials is considered. Finally, applications to Genocchi numbers $G_n$, Euler numbers $E_n$ and zeta function $\zeta(n)$ are discussed.