Mathematica Bohemica, first online, pp. 1-10


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

Emil Daniel Schwab, Gabriela Schwab

Received July 18, 2022.   Published online April 11, 2023.

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
Classification MSC:  11B39, 11A25

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Affiliations:   Emil Daniel Schwab (corresponding author), The University of Texas at El Paso, 500 West University Avenue, El Paso, Texas 79968, USA, e-mail: eschwab@utep.edu; Gabriela Schwab, El Paso Community College, 9050 Viscount Blvd., El Paso, Texas 79902, USA, e-mail: gschwab@epcc.edu


 
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