LINEAR RECURSIVE RELATIONS FOR BERNOULLI NUMBERS AND APPLICATIONS

Print ISSN: 2319-1023 | Online ISSN: 2582-5461

Abstract

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.

Keywords and Phrases

Linear recursive relations of infinite order, $\infty$-generalized Fibonacci sequences, Bernoulli numbers, Combinatorial formula, Approximation processes, Genocchi numbers, Euler numbers, zeta function, partial Bell polynomials.

A.M.S. subject classification

11B83, 11B37, 11B68.

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