FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
* L. CARLiTZ and RICHARD SCOVILLE
Duke University, Durham, Worth Carolina
and
V. £. HOGGATT,JR.
San Jose State College, San Jose, California
1.
INTRODUCTION
Let N ^ 2 be a fixed integer. We wish to discuss various properties
of sequences {v } (n = 09 ±1, ±29 -•-) of complex numbers satisfying the
recurrence
{1A)
V
n+N
=
V-N-l
+
•••
+ v
n+l
+ v
(n
n
= °> * > ±2> * ' " >
We let IT be the set of sequences satisfying (1.1) and we let B) be the set
of all sequences 5
(n = 0, ±1, ±2% • . .) which are non-zero on only a fin-
ite number of coordinates.
For 5 G B
and v G W we define
6(v) = E 8 n v n
We will call 5 6 B
(1.2)
.
canonical if
8. t 0 =£> 8. = 1
(i = 0, ±1, •••)
and
S.5 i+ 1 • • • 5 i + N - l = °
(1.3)
a = 0 , ± 1 , •••) •
We will say € and €? G D are equivalent (€ = €!) if €(v) = €'(v)
for all
v GR.
We shall also have occasion to use the translation operator
quences from W or W defined by
(1.4)
(Tv) n = v n + 1
(v G D or V) .
* Supported in part by NSF Grant GP-17031.
71
T on s e -
72
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
[jan.
The main theorem of the present paper is the following.
Theorem A. Let € E J© have integral coordinates. Then either
€ or
-€ is equivalent to a canonical element of H.
We use this theorem first to generalize a result of Kl&rner's [4] for
Fibonacci numbers to N
order Fibonacci numbers P = {P } defined by
(i)
P E W
P
W)
- ( N - 2 ) = •••
= Po = 0,
Pi = 1 .
The generalization is as follows:
Theorem B. Let Kj,
• • • ,, KI N be positive integers.
K ls K2, •••
is a unique canonical 5 E II such that
K. = SCT1?)
(1.5)
Then there
<i = 1, 2, . . . , N)
If y is a root of
/i r\
N-l
N
(1.6)
x
we let 7
De t n e
- x
-
A
-•••-. x - 1 = 0
sequence in W defined by
(7)n = 7 n •
(1.7)
We let a be the largest positive root of (1.6). Note that a > 1.
As a corollary to the main theorem we get
Theorem C. A positive real number x is of the form 5(£) for some
canonical 5 E H> if and only if, for some positive k and some integers Q l9
Q2> • " » Q N
(1.8)
we
have
akx
= Q t + Q2<^ + . . . +Q a N _ 1 .
In Section 4, we assume that N = 3 and verify some conjectures of
Hoggatt concerning certain functions introduced and discussed i n [ l ] , [2] and
1972]
[3],
FIBONACCI REPEESENTATIONS OF HIGHER ORDER — II
73
The a u t h o r s believe that the r e s u l t s obtained in Section 4 for the c a s e
N = 3 a r e strongly indicative of those that might hold for l a r g e r v a l u e s of
N.
2.
P R O P E R T I E S O F CANONICAL E L E M E N T S
T h e o r e m 1.
Suppose 6
and € G H
a r e canonical.
Then
either
6 - € o r € - 5 is equivalent to y G U .
Proof.
The n o n - z e r o c o o r d i n a t e s of rj = 5 - € a r e
l's
and - l ' s .
Suppose the f i r s t n o n - z e r o coordinate of Tj {starting from the left) i s - 1 ,
and l e t 77. = 1 be the f i r s t
^k-1
5
^k-2
3
8 5
° '
T
^k-N*
1.
^
e
Now change rj
r e s u
^
m
to 0 and add 1 to each of
S sequence i s equivalent to 77,
and
?
since 6 and € a r e canonical, it can be seen that not all of ? k _ 1 + 1, •• • »
77, _ N + 1 a r e 0.
P e r f o r m i n g this " c h a n g e " r e p e a t e d l y , we finally come to
a sequence 7f equivalent to 77 all of whose n o n - z e r o c o o r d i n a t e s a r e e i t h e r
1 o r - 1 . T h i s of c o u r s e i m p l i e s t h a t e i t h e r 77 o r
-77 i s equivalent to a
canonical e l e m e n t of D .
T h e o r e m 2. L e t € G H have i n t e g r a l c o o r d i n a t e s . Then e i t h e r c o r
-€ i s equivalent to a canonical e l e m e n t of B .
If the c o o r d i n a t e s of
€ are
non-negative then € i s equivalent to a canonical e l e m e n t of D .
Proof.
We s e t € = € - e".
The p r e v i o u s t h e o r e m shows that the
f i r s t s t a t e m e n t of the p r e s e n t t h e o r e m follows from the second; so we a s s u m e
+
€ = € .
Now a s i m p l e induction shows that it i s enough to p r o v e the following
statement:
If
€ i s canonical, then € + x. i s equivalent to a canonical e l e -
m e n t , w h e r e x- i s defined by
(2.1)
X.(V) = v.
v G V
Note that € +Xi = € - X i _ 1 - • • • - X ^ ^
T h e o r e m 1 e i t h e r yt
by T h e o r e m 1,
o r -yt
7- + X- -1
is
Is canonical.
.
+ X i + 1 = 7X
If -yt
+
X i + 1 w h e r e , by
i s canonical, then again
equivalent to a canonical e l e m e n t .
m a y suppose y± i s canonical.
Then w e get
*i ^ 1
J
+
Xi+1
s
Y
2
+
*i+2
Hence we
74
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
with yl9 y 2 j • • • canonical.
(2.2)
[€ + *.](£) >
[Jan.
But this is impossible for, if so, we would have
X . +n (£)
= a1+Ii
(n = 1, 2, •-•) .
This completes the proof.
Let P E f
(2 3)
be the sequence defined by the initial conditions
P
-
-CN-2) = ••• = Po = 0;
Pt = 1 .
Theorem 3. Let K be a positive integer. Then there is a unique canonical 5 E II such that, for all n,
(2.4)
PK=
V6.P.±
n
"
I
.
l+n
i
Proof.
Let € E H be the sequence
€ = /K
n
\0
(2 5)
^
n
=
°
otherwise
Then by Theorem 2 there is a unique canonical 5 G B
(2.6)
c(v) = 6(v),
v E ¥
satisfying
.
Letting v be translates of P we get (2.4) immediately since
c(v) = v0K
for any v E V.
The uniqueness of 8 will follow if we can show that any y E W is d e termined by its value on translates of P.
We state this as a separate
theorem.
Theorem 4. W is N-dimensional as a complex vector space.
N-l
spanned by P, TP, • • • , T
P. Moreover, the N X N matrix
A. = |(T j P) n j
(j = 0, 1, . . . , N - l )
(n = 0, i + 1, ••• , i + N - 1)
It is
1972]
FIBONACCI REPRESENTATIONS OF HIGHER ORDER — II
75
has determinant
a+i
IM - f ^ T •
Proof. The fact that V is N-dimensional is well-known, so the calculation of the determinant will complete the proof: we have
P
P.
i
p
i+
P
i
P
i+1
•i+2
i+n-l
' l+n
P
i+n-1
i+2n-2
Adding the last N - 1 columns to the first, using the recurrence and interchanging columns we get
(2.7)
\ \ \
= <-i> N + 1 !* 1 + 1 l '•
But
-(N-2)
0
0
0
0
1
0
0
0
1
1
0
1
1
1
•
.
so that
A
-(N-2)I = <-*>
N+l
Hence
IA
.i • (<-» N T
Theorem 5. Let v E V.
Then
76
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
[Jan.
(2.8) v = v 0 TP + (Vi - v0)P + (v2 - v t - v 0 ) T - 1 + . . .
+ ( v ^ + . . . - vj - v 0 )T- ( N " 2 ) P .
Proof. Let 0 s j < N - 1, The j
coordinate of the right side is
v 0 P j + 1 + (vt + v 0 ) P j + ••• + ( v N - 1
Vl
= v0(Pj+1-P.
+
+
P._ ( N _ 2 ) )
vl(P. - P ^
P._ ( N _ 2 ) )
\(PJ+l-k
+ V
P
N-2 ( P j-(N-2)+l "
+ V
- VoJP.^^
HN-2))
P
j-(N-2))
N-l ( P j-CN-2) )
The coefficient of v. is non-zero only when j + 1 - k = 1, i. e. , only when
k = j . In this case it is 1.
We can generalize a theorem proved by Klarner for the Fibonacci numbers as follows.
Theorem 6.
Let K l9 K2, K3, • • • , K N
be positive integers.
Then
there is a unique canonical 6 such that
K. = 8(TjiP)
(2.10)
Proof.
(2.11)
(i = 1, 2, ••••, N) .
It will be enough to find a canonical 8 satisfying
K. = S j T 1 " 0 ^ " 1 ^ ]
(i = 1, 2, • • • , N)
because then a translate of 8 will satisfy (2.10).
roots o f x N - x N - l - • • • - x - l = 0, and let
(2.12)
v = y
Let y be one of the N
.
Then by the previous theorem, if 8 exists and satisfies (2.11) it must also
satisfy
1972]
FIBONACCI EEPEESENTATIONS OF HIGHER ORDER - II
+ ••• + ( r N _ 1 - y N " 2 - .
S(y) = K N + (y - D K ^
N-l
1 + y + ... +y
yN
(2.13)
N-2
1 + y + . . . +y"
yN-l
^N
= K . T y - N + (KXT + KA . T 1 ) y - ( N " 1 ) + . . .
N
N
N-l'
+(
KM +
N
e B e +
K1)r-19
Hence we should define 6 to be the unique canonical form in D
to |3 E D
77
equivalent
w h e r e j3 i s given by
(2.14)
K N + . . . + K.
(-N<i<-1)
0
(otherwise)
h =
Now
/
(2.15)
N
\
pY[ TTi-(N-l)
^ - ^ P \ ) = >£J (KN
p =
(
N
+
• • • + K. J P . ^ . ^ D
/ t
- 2X E p .•j+i-(N-l) I
t=l
3.
"
K
i
Vi=l
FURTHER APPLICATIONS O F THE MAIN THEOREM
We r e c a l l that a i s the l a r g e s t positive root of
x
N
- x
N-l
x - 1 = 0
and
a =(..., a
T h e o r e m 7.
Let
K
, 1, a, • • • )
be any positive i n t e g e r .
unique canonical 5 E 11 such that
Then t h e r e e x i s t s
a
78
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
[jan.
K = 5(£) .
Moreover,
K = 8<(P) .
Proof. Choose 5 as In Theorem 3. Then
8(£) = €(£) = K .
Theorem 8. A positive real number x is of the form
£>(a) for some
canonical 6 E W if and only if, for some positive k and some integers Q l9
Q2> ••• > Q N we have
(3.1)
akx
= Qi + Q2 a + • • • + Q cF'1
.
Proof. Suppose first that x is of the form 8(a):
x =
(3.2)
€
j^
S
j=-k
Then
k
a x ~
and powers of a higher than, a N - l
\^.
i+k
~~
can be successively reduced to lower
powers eventually giving (3.1).
Now suppose (3.1) holds. Let € E M be defined by
(3.3)
€ =<
n
I
Q . ,
n k+1
t
0
- k < n < N - k - l
.
f.
otherwise
1972]
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
Then e i t h e r
€ or
79
- 6 i s equivalent to a canonical e l e m e n t 5 E 1 ,
But
€(a) = x > 0 .
Hence we m u s t have € = 8*
4 .
F o r the notation used in the r e m a i n d e r of the p a p e r we r e f e r the r e a d e r
to [3].
L e t v (M) denote the n u m b e r of n u m b e r s n E C,
T h e o r e m 9.
such that n ^ M.
If M $ C 2 then
(4.1)
v2(M)
= M - f(M) .
M o r e gene r a l l y 5 if
M $ C 2 U C 3 U •••
U
Cr
then
(4.2)
i>r(M) = f r " 2 ( M ) - f ^ M )
Proof.
Let
Kr
and let Kt = W.
to 1»
(4.3)
(r = 2, 3, 4, • • • ) .
= {K(^K $
C2 U C3 U . . .
Then c l e a r l y f
r-1
U Cr} ,
r
>2
i s 1 - 1,
onto and monotone from
M} = f r " 1 ( M )
(r = 1, 2, • • •) .
In particular.,
cardJKJK E K ^
K ^
Hence
v (M) = c a r d { K | K
E C ; K < M} = c a r d { K ( K E K r _ 1 ?
- c a r d {K|K E
K <
F ., K < M} = fr-2(M) - f
r_1
M}
(M).
Kr
80
FIBONACCI REPRESENTATIONS OF HIGHER ORDER - II
[Jan.
The following theorem is an immediate corollary.
Theorem 10. We have
(4.4)
P2(G ) = G - G 1 = G Q + G Q
^ n
n
n-1
n-2
n-3
(n > 3)
More generally
(4.5)
vr (Gn ) = Gn-r+2
^ - Gn-r+1
^ = Gn - r + G n - r - ,1
(n s> r + 1) .
Theorem 11. Let k and r be fixed integers, k ^ 1, r ^ 2. Then
(4.6)
v (kG ) = k(G
+ G
-)
r' n
n-r
n-r-1
for n sufficiently large.
Proof. Using Theorem 3, we let 6 E 11 be canonical such that
(4.7)
kG
=
n
YlS.G.^ ,
x
-'
i
(n = 0, 1, 2, •••) .
i+n
'
'
'
Hence for n sufficiently large we will have
k G n S C2 U . . .
U Cr ,
so
v (kG ) = f r " 2 (kG ) - fr""1(kG )
r
n
n
n
= S5iGi+n_(r_2) ~^5iGi+n-(r-l)
=
kG
n-(r-2) ~
kG
n-(r-l)
= k(Gn - r + G n - r - -1 ) .
The last three theorems were conjectured by Hoggatt.
REFERENCES
1. L. Carlitz, V. E. Hoggatt, J r . , and Richard Scoville, "Fibonacci Repres e n t a t i o n s / ' F ^ b o j n ^ c d ^ u a r t e r l j , Vol. 10 (1972), pp. 1-28.
[Continued on page 94. ]