Related papers: Series and sums involving the floor function
Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erd\"os and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is…
We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.
Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…
We have studied several generalizations of Fibonacci sequences as the sequences with arbitrary initial values, the addition of two and more Fibonacci subsequences and Fibonacci polynomials with arbitrary bases. For Fibonacci numbers with…
For a fixed integer N, and fixed numbers b_1,...,b_N, we consider sequences, the nth term (a_n) of which is the sum of the squares of the terms in the expansion of (b_1 + ... + b_N)^n. In the case all b_i=1, we give a formula for a…
We represent the sums $\sum_{k=0}^{n-1}{n \choose k}^{-2}$, $\sum_{k=0}^m{m\choose k}^{-1}{a\choose n-k}^{-1}$, $\sum_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\genfrac{[}{]}{0pt}{}{n}{k}}_q}$, and the sum of the reciprocals of the summands in Dixon's…
We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.
The sums of floor functions have been studied by Jacobsthal, Carlitz, Grimson, and Tverberg. More recently, Onphaeng and Pongsriiam proved some sharp upper and lower bounds for the sums of Jacobsthal and Tverberg. In this paper, we devise…
For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for…
We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these…
Motivated by questions on the ranges of commutators of dilated floor functions and one posed in Problem 27327 from Gazeta Matematic\u{a}, we investigate the precise ranges of certain generalized polynomials depending on a real parameter and…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial…
The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…
Infinite series of the type Sum{n=1,infinity}(alpha/2)_n_2F_1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced…
Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ denotes the greatest integer…
Some identities are presented that generalize the formula x^3 = 3x floor(x floor(x)) - 3 floor(x) floor(x floor(x)) + floor(x)^3 + 3 frac(x) frac(x floor(x)) + frac(x)^3 to a representation of the product x_0x_1 ... x_{n-1}.
We use an elementary argument to prove some finite sums involving expressions of the forms $(q)_n$ and $(a;q)_n$ along with inductive formulas for some sequences.