Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

On sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers on sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers

New results about some sums s _n (k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s _n (k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.     

The ring lemma in three dimensions

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c _n (R ²) = (F _2n−3−1)⁻¹—where F _k denotes the kth Fibonacci number—and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration—thought to be extremal in three dimensions—is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n − 1.     

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.      

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

Incomplete Fibonacci and Lucas numbers

A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersF _n(k) andL _n(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofL _n(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught.     

On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices

In this paper we consider certain generalizations of the well-known Fibonacci and Lucas numbers, the generalized Fibonacci and Lucas p-numbers. We give relationships between the generalized Fibonacci p-numbers, F_p(n), and their sums, , and the 1-factors of a class of bipartite graphs. Further we determine certain matrices whose permanents generate the Lucas p-numbers and their sums.     

Generating functions of the incomplete Fibonacci and Lucas numbers

For the incomplete Fibonacci and incomplete Lucas numbers, which were introduced and studied recently by P. Filliponi [Rend. Circ. Math. Palermo (2)45 (1996), 37–56], the authors derive two classes of generating functions in terms of the familiar Fibonacci and Lucas numbers, respectively.     

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let A_n=Circ(F₁,F₂,…,F_n) and B_n=Circ(L₁,L₂,…,L_n) be circulant matrices, where F_n is the Fibonacci number and L_n is the Lucas number. We prove that A_n is invertible for n > 2, and B_n is invertible for any positive integer n. Afterwards, the values of the determinants of matrices A_n and B_n can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices A_n and B_n are derived.     

On the growth of an entire dirichlet series

We establish the relation between the increase of the quantityM(σ,F) = ∣a ₀∣ + ∑ _n=1 ^∞ ∣a _n ∣ exp (σλ _n ) and the behavior of sequences (|a _n |) and (λ _n ), where (λ _n ) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a ₀ + ∑ _n=1 ^∞ a _n e ^sλn ,s=σ+it, is the Dirichlet entire series. Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 51, No. 8, pp. 1149–1153, August, 1999.     

First Observations on Prefab Posets’ Whitney Numbers

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers’ triangular array — are then calculated and the explicit formula for them is provided. Next — in the second construction — we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F-sequence independent while the second partial order is F-sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F-determined cobweb poset’s Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section. Presentation (November 2006) at the Gian-Carlo Rota Polish Seminar .     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

A generalization of Fibonacci and Lucas matrices

We define the matrix of type s, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix , whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (s=0 and s=1). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix is derived. In partial case we get the inverse of the generalized Fibonacci matrix and later known results from [Gwang-Yeon Lee, Jin-Soo Kim, Sang-Gu Lee, Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices, Fibonacci Quart. 40 (2002) 203–211; P. Staˇnicaˇ, Cholesky factorizations of matrices associated with r-order recurrent sequences, Electron. J. Combin. Number Theory 5 (2) (2005) #A16] and [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)]. Correlations between the matrices , and the generalized Pascal matrices are considered. In the case a=0,b=1 we get known result for Fibonacci matrices [Gwang-Yeon Lee, Jin-Soo Kim, Seong-Hoon Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527–534]. Analogous result for Lucas matrices, originated in [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)], can be derived in the partial case a=2,b=1. Some combinatorial identities involving generalized Fibonacci numbers are derived.     

A problem of Erdős on abelian groups

The following theorem is proved. Ifa ₁,a ₂, ...a _n are nonzero elements inZ _n , and are not all equal, then ε₁ a ₁+ε₂ a ₂+...+ε _n a _n =0 has at leastn solutions with ε _i =0 or 1.     

An approximation property of lacunary sequences

Let f ₁ and f ₂ be two positive numbers of the field , and let f _n+2 = f _n+1 + f _n for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf _n } > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf _n } < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a ⁿ } 6 min 2/(a − 1), (34a ² − 32a + 7)/(9(2a − 1)²) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa ⁿ }, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ.     

Long range variations on the Fibonacci universal code

Fibonacci coding is based on Fibonacci numbers and was defined by Apostolico and Fraenkel (1987) [1]. Fibonacci numbers are generated by the recurrence relation Fi=Fi−1+Fi−2∀i?2 with initial terms F₀=1, F₁=1. Variations on the Fibonacci coding are used in source coding as well as in cryptography. In this paper, we have extended the table given by Thomas [8]. We have found that there is no Gopala-Hemachandra code for a particular positive integer n and for a particular value of a∈Z. We conclude that for n=1,2,3,4, Gopala-Hemachandra code exists for a=−2,−3,…,−20. Also, for 1?n?100, there is at most m consecutive not available (N/A) Gopala-Hemachandra code in GH−(4+m) column where 1?m?16. And, for 1?n?100, as m increases the availability of Gopala-Hemachandra code decreases in GH−(4+m) column where 1?m?16.     

Elasticity in certain block monoids via the Euclidean table

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ _n , S) where S = (hereafter denoted ℬ _a (n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ _a (n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ _a (n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ _a (n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ _a (p). Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

Bivariate Fibonacci polynomials of order <Emphasis Type="Italic">k</Emphasis> with statistical applications

In the present article, we investigate the properties of bivariate Fibonacci polynomials of order k in terms of the generating functions. For k and ℓ (1 ≤ ℓ ≤ k − 1), the relationship between the bivariate Fibonacci polynomials of order k and the bivariate Fibonacci polynomials of order ℓ is elucidated. Lucas polynomials of order k are considered. We also reveal the relationship between Lucas polynomials of order k and Lucas polynomials of order ℓ. The present work extends several properties of Fibonacci and Lucas polynomials of order k, which will lead us a new type of geneses of these polynomials. We point out that Fibonacci and Lucas polynomials of order k are closely related to distributions of order k and show that the distributions possess properties analogous to the bivariate Fibonacci and Lucas polynomials of order k.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Spanning forests and the golden ratio

For a graph G, let f_ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_ij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(f_ij)_n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ^-1 or to 1-φ^-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.