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Keywords:
Fibonacci sequence; multiplicative arithmetic function; Binet's formula; Busche-Ramanujan identities; Möbius inversion
Fibonacci sequence; multiplicative arithmetic function; Binet's formula; Busche-Ramanujan identities; Möbius inversion
Summary:
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.
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