Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

Algebraic independence of reciprocal sums of binary recurrences

Algebraic independence of the numbers , where{R _n } _n 0 is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler's method.     

Irrationality results for reciprocal sums of certain Lucas numbers

Supported by a Habilitation Fellowship of the Deutsche Forschungsgemeinschaft.     

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Irrationality measures are given for the values of the series , where and W_n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers

Fibonacci numbers that are not sums of two prime powers

In this paper, we construct an infinite arithmetic progression of positive integers such that if , then the th Fibonacci number is not a sum of two prime powers.

     

Notes on superstrings and the infinite sums of Fibonacci and Lucas numbers

Based on infinite sums of Fibonacci and Lucas numbers, a heuristic derivation of the dimensionality of heterotic superstrings is presented. Connections to quantum chaos are briefly explored.     

h-analogue of Fibonacci numbers

We introduce the \(h\) -analogue of Fibonacci numbers for non-commutative \(h\) -plane. For \(h h'= 1\) and \(h = 0\) , these are just the usual Fibonacci numbers as it should be. We show that the Laplace integral transforms for both the Fibonacci and Chebyshev polynomials are the \(h\) -Fibonacci numbers. We also derive a collection of identities for these numbers.     

广义Fibonacci数的一些恒等式

利用非数学归纳法,以及广义Fibonacci数的性质,得到了广义Fibonacci数的一些求和公式.     

Infinite sums for Fibonacci polynomials and Lucas polynomials

The Ramanujan Journal - In this paper we establish certain infinite sums involving many arithmetical functions and the Fibonacci polynomials or the Lucas polynomials. Several of the sums are given...     

Three-term relations for Hardy sums

L. A. Goldberg (thesis, Univ. of Illinois, Urbana, 1981) discovered some three-term and mixed three-term relations for Hardy sums. His proofs are based on Berndt's transformation formulae for the logarithms of the classical theta functions. In this paper, we give elementary proofs for all of Goldberg's results and also prove some new three-term relations for Dedekind sums.     

On sums of sums involving cube-full numbers

The Ramanujan Journal - Let $$f_{3}$$ denote the characteristic function of cube-full numbers, and let (n, q) be the greatest common divisor of positive integers n and q. For any positive...     

Automatic proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

The Ramanujan Journal - Recently, Sun posed a number of conjectures on the relations between sums of squares and sums of triangular numbers. Some of these conjectures were confirmed by Baruah,...     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Relations among sums of reciprocal powers

Some formulas relating different classical sums of reciprocal powers are derived. These relations can be written in a very compact way by means of certain numbers which include Catalan’s constant and satisfy simple summation formulas. This paper has been partially supported by a DGESIC grant PB97-0342.     

Formula for Fibonacci sequence with arbitrary initial numbers

In this paper the formula for Fibonacci sequences with arbitrary initial numbers has been established by using damped oscillation equation. The formula has an exponential and an oscillatory part, it does not separate the indexes of odd and even members of the series and it is applicable on the continual domain. With elementary conditions the formula is reduced to Lucas series, and the square of Lucas series has a catalytic role in the relation of hyperbolic and trigonometric cosine. A complex function is given and the length of Fibonacci spiral is calculated. Natural phenomena support the validity of the proposed concept.     

Common factors of shifted Fibonacci numbers

For any positive integer n let F_n be the n-th Fibonacci number. Given positive integers a and b, we study the size of the greatest common divisor of F_n + a and F_m + b for varying positive integers m and n.