Fibonacci periods and multiples

2018 ◽  
Vol 102 (553) ◽  
pp. 63-76
Author(s):  
G. J. O. Jameson
Keyword(s):  
Recurrence Relation ◽  
Fibonacci Numbers ◽  
Starting Values ◽  
Image Position

The well-known Fibonacci numbers Fn are defined by the recurrence relationFn = Fn – 1 + Fn – 2. (1)together with the starting values F0 = 0, F1 = 1, or equivalently F1 = F2 = 1.We record the first few:The recurrence relation can also be applied backwards in the form Fn = Fn + 2 – Fn + 1 to define Fn for n < 0. An easy induction verifies that F−n = (−1)n – 1Fn for n > 0.

Divisibility properties for Fibonacci and related numbers

2013 ◽  
Vol 97 (540) ◽  
pp. 461-464
Author(s):  
Jawad Sadek ◽  
Russell Euler
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Unified Approach ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Image Position ◽  
Divisibility Properties

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.For non-negative integers n, the recurrence relation defined bywith initial conditionscan be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

Comment on “The Load Distribution on a Polynomial Mean Line and the Converse”

1964 ◽  
Vol 68 (637) ◽  
pp. 59-59 ◽  
Author(s):  
E. Angus Boyd
Keyword(s):  
Recurrence Relation ◽  
Load Distribution ◽  
Image Position

I am grateful to Mr. Llewelyn for pointing out the slip in equation (3) of reference 1.In considering a particular case of the polynomial camber line I did have to evaluate a number of integrals of the kind but the process is not as laborious as perhaps Mr. Llewelyn implies for there is a simple recurrence relation between integrals of this kind.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1047
Author(s):  
Pavel Trojovský ◽  
Štěpán Hubálovský
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Recursive Relation ◽  
Initial Values ◽  
Lucas Sequence ◽  
Diophantine Problems ◽  
Positive Integers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

scholarly journals On the positive roots of an equation involving a Bessel function

1989 ◽  
Vol 32 (1) ◽  
pp. 157-164 ◽  
Author(s):  
Siegfried H. Lehnigk
Keyword(s):  
Bessel Function ◽  
Recurrence Relation ◽  
Modified Bessel Function ◽  
Positive Roots ◽  
Image Position

In this paper we shall discuss the positive roots of the equationwhere Iq is the modified Bessel function of the first kind. By means of a recurrence relation for Iq(r) [2, (5.7.9)], equation (1.1a) can also be written in the form

scholarly journals Irrationality proofs à la Hermite

2011 ◽  
Vol 95 (534) ◽  
pp. 407-413
Author(s):  
Li Zhou
Keyword(s):  
Recurrence Relation ◽  
Alternative Proof ◽  
Image Position

In [1] Niven used the integralto give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's.Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].

On some infinite integrals involving logarithmic exponential and powers

1992 ◽  
Vol 122 (1-2) ◽  
pp. 11-15 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Recurrence Relation ◽  
Integral Function ◽  
Special Cases ◽  
Infinite Integral ◽  
Image Position

SynopsisIn this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ byIt is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integralSome special cases of the result are also discussed.

scholarly journals On linear recurrence sequences with polynomial coefficients

1996 ◽  
Vol 38 (2) ◽  
pp. 147-155 ◽  
Author(s):  
A. J. van der Poorten ◽  
I. E. Shparlinski
Keyword(s):  
Recurrence Relation ◽  
Upper Bound ◽  
Rational Numbers ◽  
Negative Integer ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Image Position ◽  
Recurrence Sequences

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

On a recurrence relation

1938 ◽  
Vol 5 (3) ◽  
pp. 151-154
Author(s):  
C. E. Walsh
Keyword(s):  
Recurrence Relation ◽  
Image Position

It is proposed here to consider the sequence un determined by the relationwhere, in particular,and initially u1 = θ1. The following is the main result to be proved.

Sign in / Sign up

Export Citation Format

Share Document