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A POWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES
V . E . Hoggatt , J r . , San Jose State Col leg e, San Jose, Ca
l i f .
and D . A . L i nd , Un ivers ity o f V i rg i n i a , Char l o
t tesv i l l e , V a .
1.
INTRODUCTION
The following hold for all integers
n
and
k:
F
n+k
~
F
k
F
n + i
+ F
k-l
F
n
F
n
+
k = k , )
F
^
+
(
F
k
F
k -
2
)
F
n
+ 1
( W k - ^ '
F
n
+
k
=
(
F
k
F
k - i
F
k -
2
/ 2
>
F
n
+
3
+
3 J + I ) / 2
f o r i n t e g r a l n . I n p a r t i c u l a r z
n
= F ^ o b e y s ( 2 . 1 ) . C a r l i t z , [ 1 , S e c t i o n
1 ] h a s
s h o w n t h a t t h e d e t e r m i n a n t
p + 1
J
n - j
D
n + r + s
( r , s = 0 , l , - , p )
h a s t h e v a l u e
D
p
= ( - l )
P ( P + 1
) (
n + 1
)
/ 2
| (
P
) . ( F ^ . - F p ) ^ 0
i=o \ J /
i m p l y i n g t h a t t h e p + 1 s e q u e n c e s { F } , {
F
P
} , , {F
P
} a r e l i n e a r l y
i n d e p e n d e n t o v e r t h e r e a l s . S i n c e e a c
h of t h e s e s e q u e n c e s o b e y s t h e ( p + 1 )
o r d e r r e c u r r e n c e r e l a t i o n ( 2 . 1 ) , t h e
y m u s t s p a n t h e s p a c e of s o l u t i o n s of ( 2 . 1 )
.
T h e r e f o r e a n e x p a n s i o n o f t h e f o r m ( 1 .
1 ) e x i s t s .
T o ev a lu a te the coe f f i c i en t s a . (k , p ) in (1 . 1
) we f i r s t pu t k 0 , l
s
, p ,
g iv i ng a . (k , p ) = 6 .
7
f o r 0 < j
?
k < p , w h e r e
6.-.
i s t h e K r o n e c k e r - d e l t a d e -
fi ne d by
r ( r + 1) /2
r=o
p + 1
r
a .( k - r , p ) = 0 ( j = 0 , l , - - - , p )
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A POWER IDENTITY FOR [Oct.
Now consid er b.(k,p) = (F, F, F , ) /F , . (F.F . " F F
> 0 0
F .
) for
j = 0, , ,p - 1, b (k,p) = I p j , together with the convention
that F
0
/F
0
= 1.
Clearly bj(k,p) =
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SECOND-ORDER RECURRENT SEQUENCES
277
a result parallel to (2.2)
3. EXTENSION TO SECOND-ORDER RECUR RENT SEQUENCES
We now gene ra l ize the resu l t of Sec tion 2 . Consider the
seco nd-o rder
l i ne a r r e c u r r e nc e r e l a t i on
(3.1) y
n + 2
=
P
y
n + 1
- q y
n
(q * 0) .
Let a and b be the roots of the auxil iary polynomial x
2
- px + q of (3.1),
Let w be any seque nce sat isfying
(3,1)
?
and define u by u = ( a
n
- b
n
) /
(a - b) if a + b , and u = na if a = b
s
so that u also sat is f ies (3.1) .
n
fml
Following [4 ] , we define the u-gen era l ized binomia l coef
fic ients I I by
[
T
u u u , ,
r
i
m l
=
_m _m -i rn^r+ i Tm l
=
r j
u
U iU
2
- - -u
r
| _ 0 j
u
Jar de n [4] has shown that the prod uct x of p sequ ence s each
obeying (3.1)
sa t i s f ies the (p + 1) ord er r ec ur r en ce re la t io
n
p + i
3.2) ^ T V l ) ^
1
) /
2
J=o
p + 1
J
^ =
If a l l of the se seq uen ces a re w , then i t fol lows that x
= w
p
obey s (3.2).
I t is o ur aim to give the corresp ond ing gen eraliz ation of
(1.1) for the
le
that
seque nce w ; that is , to show the re exi sts coefficients a
.(k,p,u) = a .(k) such
(3
'
3)
^
+
k
=
I ]
a
3
( k ) w J
P
n+j
j=o
and to give an explici t form for the a .(k)
e
Car l i tz [ 1 , Sec t ion 3 ] proved tha t
V
W ) =
K + r + s l
