A note on $(a,b)$-Fibonacci sequences and specially multiplicative arithmetic functions

Abstract

A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely $(a,b)$-Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly.

Keywords

• fibonacci sequence
• multiplicative arithmetic function
• binet's formula
• busche-ramanujan identities
• möbius inversion