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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.
References:
[1] Apostol, T. M.: Möbius functions of order $k$. Pac. J. Math. 32 (1970), 21-27. DOI 10.2140/pjm.1970.32.21 | MR 0253999 | Zbl 0188.34101
[2] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York (1976). DOI 10.1007/978-1-4757-5579-4 | MR 0434929 | Zbl 0335.10001
[3] Brown, T. C., Hsu, L. C., Wang, J., Shiue, P. J.-S.: On a certain kind of generalized number-theoretical Möbius function. Math. Sci. 25 (2000), 72-77. MR 1807073 | Zbl 0965.11002
[4] Catarino, P.: On some identities and generating functions for $k$-Pell numbers. Int. J. Math. Anal., Ruse 7 (2013), 1877-1884. DOI 10.12988/ijma.2013.35131 | MR 3084999 | Zbl 1285.11024
[5] Cohen, E.: Arithmetical notes. X: A class of totients. Proc. Am. Math. Soc. 15 (1964), 534-539. DOI 10.1090/S0002-9939-1964-0166146-9 | MR 0166146 | Zbl 0124.02704
[6] Falcón, S., Plaza, Á.: The $k$-Fibonacci sequence and the Pascal 2-triangle. Chaos Solitons Fractals 33 (2007), 38-49. DOI 10.1016/j.chaos.2006.10.022 | MR 2301846 | Zbl 1152.11308
[7] Haukkanen, P.: A note on specially multiplicative arithmetic functions. Fibonacci Q. 26 (1988), 325-327. MR 0967651 | Zbl 0662.10003
[8] Horadam, A. F.: Basic properties of a certain generalized sequence of numbers. Fibonacci Q. 3 (1965), 161-176. MR 0186615 | Zbl 0131.04103
[9] Horadam, A. F.: Generating functions for powers of certain generalized sequences of numbers. Duke Math. J. 32 (1965), 437-446. DOI 10.1215/S0012-7094-65-03244-8 | MR 0177975 | Zbl 0131.04104
[10] Hsu, L. C.: A difference-operational approach to the Möbius inversion formulas. Fibonacci Q. 33 (1995), 169-173. MR 1329025 | Zbl 0822.11005
[11] Jhala, D., Sisodiya, K., Rathore, G. P. S.: On some identities for $k$-Jacobsthal numbers. Int. J. Math. Anal., Ruse 7 (2013), 551-556. DOI 10.12988/ijma.2013.13052 | MR 3004318 | Zbl 1283.11028
[12] Kalman, D., Mena, R.: The Fibonacci numbers---exposed. Math. Mag. 76 (2003), 167-181. DOI 10.2307/3219318 | MR 2083847 | Zbl 1048.11014
[13] McCarthy, P. J.: Introduction to Arithmetical Functions. Universitext. Springer, New York (1986). DOI 10.1007/978-1-4613-8620-9 | MR 0815514 | Zbl 0591.10003
[14] McCarthy, P. J., Sivaramakrishnan, R.: Generalized Fibonacci sequences via arithmetical functions. Fibonacci Q. 28 (1990), 363-370. MR 1077503 | Zbl 0721.11008
[15] Sastry, K. P. R.: On the generalized type Möbius functions. Math. Stud. 31 (1963), 85-88. MR 0166140 | Zbl 0119.28001
[16] Schwab, E. D., Schwab, G.: $k$-Fibonacci numbers and Möbius functions. Integers 22 (2022), Article ID A64, 11 pages. MR 4451564 | Zbl 07569238
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