A FIBONACCI CIRCULANT
D. A. LIND
University of Virginia, Charlottesville, Virginia
1. Put
1 F
r
F
F
D
n,r
F
r+n-1
F
F
r+n-2
r+2
'r+n-1
r+1
"r+n-2
F
r+n-1
F
' Fr+1
where F
F
r+1
F
r
F
r+2
r
r+n-3
r+3
denotes the Fibonacci numbers defined by
J
n
Yi = Fo = 1.
1
2
*
F
= F
n+2
+ F
n+1
n
We show that
(F
(1)
D
n,r
(F
- F
)
1
;
"n+r
n+r-1
r
l
1
1 - L + (-1) "
n
where L
= F - + F , - is the n
Lucas number*
n
n-1
n+1
A circulant is a determinant of the form
a
a
0
a
n-1
(2)
C(a0,"- ' a n - l } =
a
n-2
l
0
Vi
a
a
2
l
a
o
S O S
Vi
see
a
s a «
an-3
0
n-2
e
e
a
a
i
449
2
a
3
9 • 9
a
o|
450
A FIBONACCI CIRCULANT
Dec.
It is known (see [1, Vol. 3 9 pp. 374-375] and [3, p . 39]) that
n-l/'n-l
. \
(3)
where the
,,
2k?r , .
.
2k7T
Xt) = cos
+ i sin
k
n
n
th
are the n roots of unity* To establish (3) rapidly s multiply
C
S
Ctao,.-..^)
bytheVandermonde determinant V = Jco?I (i s j = 0S 1, 8 8 ° , n - 1). Denoting
the right side of (3) by P t by factoring out common factors, one finds CV =
PV, and since V ^ 0, (3) follows.
Now D
is a special case of (2) with
n9 r
a. = F . ,
j
j+r
J+r
in which a = (1 + N / 5 ) / 2 , /3 = (1 -y/E)/2.
- fl>+r
a - 0
Thus by (3),
Ji+r +
, r[ar - 1 - ,jc-l
ji+v-1-.Jo,
n-1- a r - /3„r - an+r +, (r
/3
-a n+r-1 ,+/S
=
k^0
n X
" k =o
" " > - 0 ( 1 - « 0 k ) ( l - /3w k )
F
r " F n + r " < F n+r-l " F r - l ) W k
- T i - <*>kHi - ^ )
1970
451
A FIBONACCI CIRCULANT
Now for any x and y ,
(4)
Ivx-y^-y°S(!-"*)-4©"
= x
n
n
- y
Therefore
£ ,
CFr " F n + r " ( F n + r - l ~
F
r-l)C\l
(F
'
r " W
"
" <Fn+r-l " V l * "
and
n-1
n-1
n-1
n a - aoj ){i - pw ) « n a - aoj ) n (i = (1 - an)(l
w h e r e we have u s e d L M = a + p
s
- ^)
JSW.
1 - L n + (-l)n
-
.
simplifies if n i s even* Ruggles
n i ic
[ 2] h a s shown that
I
F _,_ - F
n+p
It follows that if n = 0 (mod 4 ) ,
L F ,
n p
p even
F L ,
n p
p
odd
,nfTn
T
F
n
-i
2 - L
and if n s 2 (mod 4) s
n r n
D
n9r
f
T h i s e s t a b l i s h e s (1).
We note that
that t h i s evaluation of D
D
>
1
vn
r 1+ in
-i
H M 2 - ~L - 2 J
452
A FIBONACCI CIRCULANT
[Dec.
2. The generalization of (1) to second-order recurring sequences uses
the same techniques. Consider the sequence { w } defined by
n+2
* n+1
^ n
WQ and Wj arbitrary, where p 2 - 4q ^ 0. Let a and b be the roots of
the auxiliary polynomial, so that a ^ b and ab = q. We shall assume that
neither a nor b is an n
root of unity* Since the roots are distinct, there
are constants A and B such that W
{V n } by
n
= A a + Bb .
Define the sequence
n
Vn = a + b .
Put
D
(W) = C(W , W ^ , ' •' , W ^ J
n,r
r
r+1
n+n-1
Setting a. = W. + r = Aa : i + r + Bb^ +r in (3) gives
Dn
n , r
n-1 / n - 1
.
A
(W) = n ( £ Aa r (ao). ) j + Bbr(bco ) ] )
K
K
k=0 \ j=0
/
- V /*a r (l - a">
k=0 \
n
n
k=0
W
1
~
r -
aw
W
+
Bbr(l - b n ) \
1
k
~
bw
k
/
n+r " ^ W r - 1 ~ W n + r - l ^ k
(1 - a c k ) d - b a y
(W r - W n + r ) n - q n (W r -l - W ^ ^ ) 1 1
1 - Vn
+
qn
which agrees with (1) by taking p = 1, q = - 1 , W = F , and V
3, We now consider a slight variant of the above.
Put
= L .
1970]
453
A FIBONACCI CIRCULANT
r+1
F
r+n-1
E
r+n-2
nfr
-F
F
r+n-1
r+1
r+n-2
F
r+n-3
-F
r+2
r+1
r+n-1
"r+2
F
r+3
We shall prove
(Er + F n+r
^_ f
(5)
n,r
+ (-l) n (F n+r-1
^ i + F r - 1 .)n
1 + L + (-l) n
n
A determinant of the form
s
V"'Vi
*n-l
)
is termed a skew circulanfc,
"an-2
"Vl
"al
~a2
a
n-2
a
n-3
Scott [1, Vol. 4, p. 356] has shown that
n-1 / n-1
. \
where the
cos
n-l
~a3
(6)
€k =
a
(2k + l)ir __, . . (2k + l)ir
1 — J - + i sin — 2
454
[Dec.
A FIBONACCI CIRCULANT
a r e the n
th
r o o t s of - 1 . To p r o v e (6) quickly, multiply S(a 0 , • • • , a
the Vandermonde d e t e r m i n a n t | e] | (i, j = 0S 1, * • • , n - 1),
1
) by
and t r e a t a s in
the proof of (3).
To evaluate E
n,r
l e t a. * F . , .
j
j+r
A development s i m i l a r to Section 1
F
shows that
n - 1 / n - 1 aT(a€, ) j - /3r(j3€, );
a - j8
k=0 \ j »=0
••)
(7)
n-1
n
F
n+r
+ F
+ TF , - + F
-If.
r
L n+r-1
r-U k
(1 - ar€k )(l - iS€k )
"
k=0
F o r a r b i t r a r y x and y ,
(8)
Iv^'-SO^M©
+ 1
x
n _, n
+ y
Application of this to (7) yields the d e s i r e d r e s u l t (5).
We r e m a r k that a s before (5) simplifies for even n.
shown that
F ^ + F
n+p
n-p
F L ,
n p
L F ,
n p
p even
^
p odd
F
Then if n s 0 (mod 4),
_ ^(*Hn
nsr
+ F
2 + L
r-l+|n)
Ruggles [ 2 j h a s
1970]
A FIBONACCI CIRCULANT
455
while if n = 2 (mod 4),
|nl
r+|n
r-l+|nj
E
2 + L
n
Note that the l a t t e r y i e l d s on c o m p a r i s o n with the d e t e r m i n a n t the identity
5(F 2 , + F 2 ) = L 2
+ L 2 = 5F Q _,_r+1
r
r+1
r
2r+l
4.
.
The extension of this to s e c o n d - o r d e r r e c u r r i n g s e q u e n c e s involves
no new i d e a s , and the d e t a i l s a r e t h e r e f o r e omitted.
and V be a s
n
th
before j with the exception that we r e q u i r e a and b not be n
r o o t s of - 1
r a t h e r than +1 to avoid division by z e r o . P u t
E
n,r
Let W
n
(W) = S(W , W ^ , • • • , W ^ , ) .
r
r+1
r+n-1
Using (6) and (8), we find
E
(W)
(W
n+r
+
W
r
) +
q
n ( W
n+r-l
1+V
n
which r e d u c e s to (5) when q = - q = 1,
+
W
r-l)n
11
W
H-q
^
= F , and V
= L
.
REFERENCES
1.
T h o m a s M u i r , The T h e o r y of D e t e r m i n a n t s (4 V o l s . ) , D o v e r , New York,
I960.
2* L D. R u g g l e s , "Some Fibonacci R e s u l t s Using F i b o n a c c i - T y p e S e q u e n c e s / '
Fibonacci Q u a r t e r l y , 1 (1963), No. 2, pp. 75-80,
30
V* I. S m i r n o v , L i n e a r A l g e b r a and Group T h e o r y , M c G r a w - H i l l , New
Y o r k , 1961.