Sums of the Fibonacci and Lucas numbers over the binary digital sums

Yıl 2018, Cilt: 6 Sayı: 2, 286 - 289, 15.10.2018

Teerapat Srichan

Öz

Let $s(n)$ denote the binary digital sum of the positive integer $n$. Using elementary combinatorial method, we present some known identities as a new form in term of $s(n)$. The sums of the Fibonacci, Lucas and harmonic numbers over the binary digital sums are considered. Moreover, we also give the sums over the binary digital sums, which derived from the binomial theorem.

Anahtar Kelimeler

binary digital sum, Fibonacci number, harmonic number, Lucas number

Kaynakça

• [1] G. Boole., Calculus of Finite Differences, 4th Edition, Chelsea, New York, 1957.
• [2] K. Dilcher., Some q-series identities related to divisor functions, Discrete mathematics, 145.1-3,(1995), 83-93.
• [3] M. Kr¨uppel., “De Rham’s singular function, its partial derivatives with respect to the parameter and binary digital sums,” Rostock. Math. Kolloq., 64, (2009), 57–74.
• [4] T. Koshy., Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 2001.
• [5] J. S´andor, and B. Crstici., Handbook of Number Theory II., Kluwer Academic, Dordrecht, 2004.
• [6] E. Trollope., “An explicit expression for binary digital sums,” Mat. Mag., 41, (1968), 21–25.