PELL IDENTITIES
A. F. HORADAIVI
University of New England, Armidale, Australia
1. INTRODUCTION
Recent issues of this Journal have contained several interesting special
results involving Pell numbers* Allowing for extension to the usual Pell numbers to negative subscripts, we define the Pell numbers by the Pell sequence
{pj thus:
(p \ . ' "
•••
1 ;
P
-4
-12
P
- 3 P-2 P - i PO Pi P 2 P 3 P4 P 5 e«
5 -2
1 0 1 2 5 12 29 .
in which
(2)
Po = 0,
Pl
= 1,
Pn+2 = 2Pn+1
+
Pn
and
(2?)
v
'
= (-l)n+1P
P
-n
.
n
The purpose of this article is to urge a greater use of the properties of
the generalized recurrence sequence ( w (a,b; p s q)} 9 discussed by the author
in a series of papers [2], [3],
and [4]„ The Pell sequence is but a special
case of the generalized sequence.
2S THE SEQUENCE {W n (a 9 b; p ? q)}
Our generalized sequence {W n (a 5 b; p 9 q)} is defined [2] as
• • • W_i 5 W 0 5 W i 5 W 2 ,
(3)
W3 ,
W4 • • •
{Wj:
...
R™L_. j a 9 b jpb-qa 9 p 2 b-pqa-qb ,• • •
245
246
PELL IDENTITIES
[May
in which
(4)
W0 = a,
Wi = b,
W n + 2 = pW n + 1 - qW n ,
where a, b s p, q are arbitrary integers at our disposal.
The Pell sequence is the special case for which
(5)
a = 09
i.e.,
b = 1,
p = 2,
q = -1,
P n = W n (0, 1; 2 - 1).
From the general term W
[2] , namely,
W = ^ - ^ - aa + 2 £ ^ b
n
a - p
a - p
(6)
H
^
'
where
a = (p + d)/2,
(7)
P = (p - d)/2
] 2
^iV
! d = (p2 - 4q)
We have, for the Pell sequence, using (5),
(8)
d = 23/2
{ a = 1 + <s/2
/S = 1 -
so that ? from (5), (6) and (8), the n
N/2
term of the Pell sequence is
*. - " + %,i(i
-
A generating function for ( w }, namely [ 4 ] ,
^
1971]
PELL IDENTITIES
a + (b
(10)
-
pa)x
1 - pz + qx
2
247
= £>nxn
n=0
becomes, using (5) for {p } s
n^
X
(11)
1 - 2x - x 4
=EWnXn
A
n=0
n=0
Associated with {w } is [2] the characteristic number
e = pab - qa2 - b2
(12)
with Pell value
(13)
ep = -1
by (5).
Another special case of subsequent interest to us in (32) is the sequence
(U n (p,q)} defined by
(14)
U0 = 0,
UA = 1,
U
+ 2
=
i.e. ,
U n (p 9 q) = W n (0 9 1; p,q) ,
for which
(15)
e n = -1
PU
- qUn ,
248
PELL IDENTITIES
[May
and
U
= -q~nU .
-n
^
n
(16)
Result (16) was noted long ago by Lucas [ 6 ] , p. 308, to whom much of the
knowledge of sequences like { u (p,q)} is due. Obviously, by (5) and (14),,
(17)
P n = U n (2, -1) .
3. PELL IDENTITIES
Specific Pell identities to which we refer are:
[(k-l)/2]
(18)
r=0
(19)
k
P
2k = Z(rK*:r
r=l
(20)
P Q . = p2 + p2
2n+l
n
n+1
(21)
P 0 ^ + P Q = 2P2
- 2P 2 - (-l) n
2n+l
2n
n+1
n
(22)
(-l) n P P. = P __, P . - P P , . .
a b
n+a n+b
n n+a+b
These identities occur as Problems B-161 [ 5 ] , B-161 [5], B-136 ,[7], B-137
[ 7 ] , and B-155 [8], respectively.
Identity (18) follows readily from formula (3.20) of [2]:
[n/2]
(23)
j=0
[(n-l)/2]
+
(2b-Pa) Y:
j=0
Lf^y-v-w
1971]
P E L L IDENTITIES
249
on using (5) and (8).
Identity (19) follows from f o r m u l a (3.19) of [ 2 ] :
on using (5) and recognizing that
r=0
r=l
Employing the formula (3.14) of [2] and r e p l a c i n g U n t h e r e i n (and s u b sequently as r e q u i r e d ) by U
(26)
- in a c c o r d a n c e with (14) to get
W^
= W U .,_- - qW - U
,
n+r
r n+1
^ r-1 n
we put r = n + 1,
and identity (20) follows i m m e d i a t e l y with the aid of (5)
and (17).
F u r t h e r m o r e s (20) m a y s i m p l y be obtained from f o r m u l a (4.5) of [ 2 ] :
(27)
W , W
= W2 + e q n " r U 2
n+r n - r
n
^
r
on choosing r = n + 1 and utilizing (1), (5), (13) and (17). (P_ 4 = P 4 = 1.)
An i m m e d i a t e consequence of (26) i s , by (5) and (17), the r e s u l t
(28)
x
7
P ,
n+r
= P P ,1 + P - P
r n+1
r-1 n
Setting r = n in (28), we deduce that
< 29 >
P
2n
= Pn^l^n-l)'
F r o m (27), with r = 1 and using (5), (13) and (17)
(30)
P
. P , - P2 =
n+1 n - 1
n
(-l)n.
(P 4 = 1),
we have
250
P E L L IDENTITIES
[May
Now, to p r o v e identity (21), m e r e l y add (20) and (29).
P
2n+1
+ P
2n
=
=
P
+
U
P
n+1
< P n+l P n - 1 " <-«*>
+ ]
WPn+l "
2
+ P
n<2Pn+l "
2P
n> " <-«*
+ P
n
Then
2P
(2P
n>
n+l "
2
V
n
= 2(P2
- - P ) - (-l)
v
n+1
n
on using (2) t w i c e , and (30).
Next, c o n s i d e r f o r m u l a (4.18) of [ 2 ] :
W W L _,_. - W W ,, = e q n " r U U ,, .
n - r n+r+t
n n+t
^
r r+t
(31)
Put r = - a , b = r + t ,
Using (2 ! ), (5), (13), and
t = a + b in (31).
(17), we o b s e r v e t h a t identity (22) evolves without difficulty.
4.
I.
CONCLUDING COMMENTS
P r o b l e m B - 1 7 4 , p r o p o s e d by Zeitlin [10] from the solution to P r o b l e m
B-155 [ 8 ] , n a m e l y , to show that
(32)
UJ.U_LU-UUJ__I.K=
n+a n+b
n n+a+b
q n U U, ,
a b
i s p r o v e d for identity (22) from (31) on using (14), (15), and (16).
II. D i s c u s s i n g briefly the sequence { T } for which
(33)
T
=
n
where
(34)
r
- 1 + \/5
2
s
=
1
-
N/5
and A , B depend on initial conditions, B r o . B r o u s s e a u
a n s w e r s , the q u e s t i o n s :
(i) Which s e q u e n c e s have a l i m i t i n g r a t i o T / T
(ii) Which s e q u e n c e s do not have a l i m i t i n g r a t i o ?
-?
[1] a s k s , and
1971]
251
P E L L IDENTITIES
(iii) On what d o e s the l i m i t i n g r a t i o d e p e n d ?
He finds that
(35)
lim
n-1
= r
T h i s a c c o r d s with o u r m o r e g e n e r a l r e s u l t (3,1) of r 2 ] :
W
(36)
lim
where
W
a
P
n-1.
<x9p a r e defined
if
if
in (7).
|j8| < 1
\a\ < 1
Result
s
(36) p r o b a b l y
answers
f
Bro* B r o u s s e a u s q u e r i e s (i), (ii) ? (iii) from a slightly different
point of view.
C l e a r l y , the p a r t i c u l a r sequence he q u o t e s , n a m e l y , the one defined by
(37)
i. e*
Ti1 = 5 ,
*
?
T 2L = 9,9
o u r { w (5,9; 3 , 4 ) } ,
T
l0
n+2
= 3T _,, - 4T
n+1
n
cannot converge to a r e a l l i m i t , since by (7),
a = (3 + i<s/7)/2
(38)
j3 =
(3 -
1N/7)/2
which a r e both complex n u m b e r s .
JUL C o r r e s p o n d i n g to the specifically stated P e l l identities (18)-(22), and to
the incidental P e l l identities (28)-(30), one m a y w r i t e down identities for the
(39)
j F i b o n a c c i sequence
(F }
= { w (0, 1; 1, -1)}
J L u c a s sequence
{L }
= ( W n ( 2 , 1; 1, -1)} .
t G e n e r a l i z e d sequence ( H (s,r)}
= { w ( s , r; 1, -1)}
R e a d e r s a r e invited to explore t h e s e p l e a s a n t m a t h e m a t i c a l p a s t u r e s *
Re-
v e r s i n g o u r p r e v i o u s p r o c e d u r e of u s i n g the { W } sequence to obtain special
252
P E L L IDENTITIES
May 1971
(Pell) i d e n t i t i e s , one could be motivated to d i s c o v e r g e n e r a l i z e d W
identities
c o m m e n c i n g with only a s i m p l e r e c u r r e n c e - r e l a t i o n r e s u l t .
C o n s i d e r , for e x a m p l e , the r e l a t i o n s h i p
F2 + F2
= 2(F 2
+ F2
) .
n
n+3
n+1
n+2
(40)
an a e s t h e t i c a l l y a t t r a c t i v e r e s u l t known in embryonic f o r m , at l e a s t , In 1929
when it w a s d e s c r i b e d in a philosophical a r t i c l e by DVArcy Thompson [9] a s
TT
another of the many c u r i o u s p r o p e r t i e s " of ( F }- R e a d i l y , we have
(41)
L2 + L2
= 2(L 2
+ L2
)
n
n+3
n+1
n+2
(42)
H 2 + H2
= 2(H2
+ H2
) .
n
n+3
n+1
n+2
Not unexpectedly, the r e s u l t s (40)-(42) a r e alike simply b e c a u s e we have
p = 1, q = - 1 for each of the s e q u e n c e s concerned.
But w h a t , we a s k , will
happen in the c a s e of the P e l l s e q u e n c e , for which p = 2 ,
q = -1?
P r o c e e d i n g to the g e n e r a l i z e d situation, we find
(43)
W^ + w 2 n + 3 = q" 2 ( p V + D W 2 ^ + (p2 + q 4 )W2 n + 1
P e l P s sequence r e d u c e s (43) to
P2 + P2
= 5(P 2 J _ 1 + P 2
) .
n
n+3
n+1
n+2
(44)
IV.
By now, the m e s s a g e of this a r t i c l e should be e v i d e n t
Simply, it i s
this:
While the d i s c o v e r y of individual p r o p e r t i e s of a p a r t i c u l a r s e q u e n c e ,
elegant though they m a y b e , i s a satisfying e x p e r i e n c e , I believe that a m o r e
fruitful m a t h e m a t i c a l e n t e r p r i s e i s an investigation of the p r o p e r t i e s of the
g e n e r a l i z e d sequence { w }.
a r e brought to light.
In this way,
o t h e r w i s e hidden
To t h i s o b j e c t i v e , I c o m m e n d the r e a d e r ,
[Continued on page 263. ]
relationships