Download A Nonlinear Expression for Fibonacci Numbers and Its Consequences

Journal of Mathematical Research with Applications
Nov., 2012, Vol. 32, No. 6, pp. 654–658
DOI:10.3770/j.issn:2095-2651.2012.06.003
Http://jmre.dlut.edu.cn
A Nonlinear Expression for Fibonacci Numbers and Its
Consequences
Leetsch C. HSU
School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
Abstract Here investigated is a kind of nonlinear combinatorial expression involving Fibonacci numbers defined on the set of integers. A number of particular consequences will be
presented as examples.
Keywords
difference operator; combinatorial identity; congruence relation.
MR(2010) Subject Classification 05A15; 05A19; 11B73
1. Introduction
First of all, it may be worth mentioning that a great many research works have been done
for Fibonacci number sequence and the like during past 60 years. The vast literature may be
found more easily in the Fibonacci Quarterly, the journal that started publication since 1963.
Denote by N and Z the natural number set (including O) and the set of integers, respectively.
It is known that Fibonacci numbers could be defined on N as well as on Z (cf. Graham-KnuthPatashnik [1]). More percisely, the two sets of numbers {Fn } (n ∈ N) and {fn } (n ∈ Z) are
generated by the following recurrence relations (with initial conditions) respectively:
F0 = F1 = 1, Fn+1 = Fn + Fn−1 , n ≥ 1,
(1)
f0 = 0, f1 = 1, fn+1 = fn + fn−1 , n ∈ Z.
(2)
Usually we also define F−1 = 0. Comparing (1) with (2), one gets the simple relation
fn = Fn−1 , n ∈ N.
(3)
Also, one may easily find by induction the relation (cf. loc. cit.)
f−n = (−1)n−1 fn , n ∈ Z.
(4)
Moreover, using the ordinary difference operator ∆ with increment 1, we have
∆ft = ft+1 − ft = ft−1 , ∆k ft = ft−k , k ∈ N, t ∈ Z.
This suggests that Difference Calculus should be most easily applied to the f -sequence {fn } (n ∈
Z).
Clearly, (3) implies that the Binet formula for fn is given by
√
fn = (an − bn )/ 5
Received May 8, 2012; Accepted September 3, 2012
(5)
A nonlinear expression for Fibonacci numbers and its consequences
where a = 12 (1 +
all n ∈ Z.
√
655
√
5) and b = 12 (1 − 5). In fact, it is easily verified via (4) that (5) holds for
In this short paper we will prove a theorem containing a combinatorial algebraic formula
for fmn+r (m, n ∈ N, r ∈ Z) which involves numerous special identities as consequences. Also,
as a further application of the theorem, several congruence relations will be displayed in the last
section (§3).
2. A theorem and its consequences
Theorem For any given m, n ∈ N and r ∈ Z there holds a combinatorial formula of the form
n X
n n−k k
fmn+r =
f
f fr+k .
(6)
k m−1 m
k=0
Corollary For m ≥ 1 and r ≥ 0 there is a formula of the form
n X
n
Fmn+r =
F n−k F k Fr+k .
k m−2 m−1
(7)
k=0
Obviously, from the view-point of algebraic computations, both (6) and (7) are utilizable when
m is much bigger than n, and provided that the values of fm−1 , fm and fr+k (0 ≤ k ≤ n) are
given or more easily determined.
Proof of Theorem We shall show that formula (6) could be obtained via a suitable application
of Difference Calculus. Particularly, use will be made of the shift operator E n = (I + ∆)n with
I denoting the identity operator, namely, If (t) ≡ f (t) for any function f (t) defined on Z.
Clearly, the simplest case n = 1 of (6) is the following identity
fm+r = fm−1 fr + fm fr+1 , r ∈ Z.
In accordance with (5), we see that (8) is equivalent to
√ m+r
5(a
− bm+r ) = (am−1 − bm−1 )(ar − br ) + (am − bm )(ar+1 − br+1 ).
Actually, this can be verified by algebraic computation and by using the facts that a + a−1 =
√
b + b−1 = − 5 and a + b−1 = a−1 + b = 0. The routine computation is omitted here.
(8)
√
5,
Replacing r by r + mt in (8), we have
fr+mt+m = fm−1 fr+mt + fm fr+1+mt .
Let F (t) = fr+mt . Then it follows that
∆F (t) = F (t + 1) − F (t) = fr+mt+m − fr+mt = fm fr+1+mt + (fm−1 − 1)fr+mt .
This may be rewritten symbolically in the form
(∆ − (fm−1 − 1)I)fr+mt = fm · fr+1+mt .
Using the operator (∆ − (fm−1 − 1)I) twice, we find
2
· fr+2+mt .
(∆ − (fm−1 − 1)I)2 fr+mt = fm
(9)
656
Leetsch C. HSU
Thus it is easily seen that by induction we may get the general expression
k
(∆ − (fm−1 − 1)I)k fr+mt = fm
· fr+k+mt , k ∈ N.
(10)
Consequently, for F (t) = fr+mt we can make use of (10) to compute F (n) as follows
F (n) = fmn+r = (I + ∆)n fmt+r t=0
= (fm−1 I + (∆ − (fm−1 − 1)I))n fmt+r t=0
n X
n n−k
=
fm−1 (∆ − (fm−1 − 1)I)k fmt+r t=0
k
k=0
n
X n
=
f n−k f k fr+k .
k m−1 m
k=0
This completes the proof of theorem. Consequence 1 For m ≥ 1 and n ≥ 1 there holds the identity
n X
n
n−k k
(−1)k Fm−2
Fm−1 Fmn−k = (−1)mn .
k
(11)
k=0
Here the simplest case given by n = 1 is the oldest theorem due to Cassini and Kepler, viz.
2
Fm−2 Fm − Fm−1
= (−1)m .
(12)
In fact, (11) is a particular case of (6) with r = −mn − 1, namely
n X
n
f−1 = f1 = 1 =
F n−k F k fk−mn−1
k m−2 m−1
k=0
mn−k
where fk−mn−1 = (−1)
fmn−k+1 = (−1)mn−k Fmn−k .
Consequence 2 For m = 1, 2, 3, 4, 5, 6, formula (6) yields the following special instances for
r ∈ Z:
n X
n
fr+k = fr+2n ,
k
k=0
n X
n k
2 fr+k = fr+3n ,
k
k=0
n X
n n−k k
2
3 fr+k = fr+4n ,
k
k=0
n X
n n−k k
3
5 fr+k = fr+5n ,
k
k=0
n X
n n−k k
5
8 fr+k = fr+6n .
k
k=0
For m ≥ 2 we may rewrite (7) in the form (with r ≥ 0)
n X
n
n
Fmn+r /Fm−2
=
(Fm−1 /Fm−2 )k Fr+k .
k
k=0
A nonlinear expression for Fibonacci numbers and its consequences
657
Clearly, this could be embedded in the δ-series transform. By the inverse transform we find
n X
n
k
(−1)k Fmk+r /Fm−2
= (−1)n (Fm−1 /Fm−2 )n Fr+n .
k
k=0
This leads to the following
Consequence 3 For given m ≥ 2 and r ≥ 0, the δ-inverse of (7) is given by
n X
n
n−k
(−1)k Fm−2
Fmk+r = (−Fm−1 )n Fr+n .
k
(13)
k=0
A number of particular cases may be displayed as follows:
n X
n
(−1)k Fr+3k = (−2)n Fr+n ,
k
k=0
n X
n
(−1)k 2n−k Fr+4k = (−3)n Fr+n ,
k
k=0
n X
n
(−1)k 3n−k Fr+5k = (−5)n Fr+n ,
k
k=0
n X
n
(−1)k 5n−k Fr+6k = (−8)n Fr+n ,
k
k=0
n X
n
(−1)k 8n−k Fr+7k = (−13)n Fr+n .
k
k=0
It is conceivable that certain special identities mentioned in Consequences 2 and 3 may be
found in the literature. Note that proof of (6) also holds for r, m ∈ Z.
3. More consequences: congruence relations
The nonlinear expressions (6) and (7) could be used to obtain some congruence relations
involving products of some Fibonacci numbers as modules. First of all, it is easily observed that
(6) implies the following
Consequence 4 For r ∈ Z, m ≥ 1 and n ≥ 1, there holds a congruence relation of the form
n
n
fmn+r ≡ fm−1
fr + fm
fr+n (mod fm−1 fm ).
(14)
In particular, setting r = 0 leads to
n
fmn ≡ fm
fn (mod fm−1 fm ) ≡ 0 (mod fm ).
(15)
fmn ≡ 0 (mod fn ).
(16)
Similarly to (15), we have
Thus, if fm and fn are relatively prime, viz. gcd(fm , fn ) = 1, then (15)–(16) imply the relation
fmn ≡ 0 (mod fm fn ).
(17)
658
Leetsch C. HSU
Generalizing the above arguments, we get the following
Consequence 5 Let fm1 , . . . , fms be relatively prime to each other with each mi ≥ 1. Then
there holds the relation
fm1 ···ms ≡ 0 (mod fm1 · · · fms ).
(18)
This is precisely equivalent to the well-known theorem:
Fm1 ···ms −1 ≡ 0 (mod Fm1 −1 · · · Fms −1 )
(19)
where all Fmi −1 (1 ≤ i ≤ s) are relatively prime to each other.
n
Setting r = −n in (14) yields f(m−1)n ≡ fm−1
− f−n (mod fm−1 fm ). Assuming n to be an
odd positive integer and replacing m by m + 1, we have
n
n
(mod fm fm+1 ).
fmn ≡ (−1)n−1 fn fm
(mod fm fm+1 ) ≡ fn fm
(20)
Comparing (20) with (15) and using the property gcd(fm−1 , fm+1 ) = 1, we find
n
fmn ≡ fn fm
(mod fm−1 fm fm+1 ).
(21)
This leads to the following
Consequence 6 There is a congruence relation of the form
n+1
F(m+1)(n+1)−1 ≡ Fn Fm
(mod Fm−1 Fm Fm+1 )
(22)
where m ≥ 1 and n is a positive even integer.
Remark In a recent paper [2; §3] the author has adopted the same denotation {Fn } (n ∈ Z) as
that used in [1; §6.6] for defining the Fibonacci number sequence on Z via the conditions
F0 = 0, F1 = 1, Fn+1 = Fn + Fn−1 , n ∈ Z.
√
However in such case Fn = (an − bn )/ 5 and the partial sequence {Fn } (n ∈ N) is not consistent
with the most commonly used denotation {Fn } for the classical Fibonacci numbers defined on
N (cf. [3] and various popular books on combinatorics). This is the reason why we now propose
using the denotation {fn } (n ∈ Z) instead of {Fn } (n ∈ Z), although for n ≥ 1 the relation
fn = Fn−1 is rather simple. Finally it may be worth expecting that formulas (6) and (7) could
be possibly extended to certain kinds of generalized Fibonacci number sequence with suitable
initial conditions.
References
[1] R. L. GRAHAM, D. E. KNUTH, O. PATASHNIK. Concrete Mathematics. Addison-Wesley Publishing
Company, Reading, MA, 1994.
[2] L. C. HSU. Concerning two formulaic classes in computational combinatorics. J. Math. Res. Appl., 2012,
32(2): 127–142.
[3] H. S. WILF. Generating Functionology. Acad. Press, New York, 1994.